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In [[mathematics]], a '''ring''' is an [[algebraic structure]] with two binary operations, commonly called ''addition'' and ''multiplication''. These operations are defined so as to emulate and generalize the [[integer]]s. Other common examples of rings include [[polynomials]] and [[matrix|matrices]].
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{{TOC|right}}
 
In [[mathematics]], a '''ring''' is an [[algebraic structure]] with two binary operations, commonly called ''addition'' and ''multiplication''. These operations are defined so as to emulate and generalize the [[integer]]s. Other common examples of rings include the ring of [[polynomial]]s of one variable with real coefficients, or a ring of square [[matrix|matrices]] of a given dimension.
 
To qualify as a ring, addition must be commutative and each element must have an inverse under addition:  for example, the additive inverse of 3 is -3.  However, multiplication in general does not satisfy these properties.  A ring in which multiplication is commutative and every element except the additive identity element (0) has a multiplicative inverse (reciprocal) is called a [[field]]: for example, the set of rational numbers.  (The only ring in which 0 has an inverse is the trivial ring of only one element.)
 
A ring can have a finite or infinite number of elements.  An example of a ring with a finite number of elements is <math>\mathbb{Z}_5</math>, the set of remainders when an integer is divided by 5, i.e. the set {0,1,2,3,4} with operations such as 4 + 4 = 3 because 8 has remainder 3 when divided by 5.  A similar ring <math>\mathbb{Z}_n</math> can be formed for other positive values of <math>n</math>.


==Formal definition==
==Formal definition==
A '''ring''' is a [[set]] ''R'' equipped with two [[binary operation]]s which are generally denoted + and · and called ''addition'' and multiplication'', respectively, such that
A '''ring''' is a [[set]] ''R'' equipped with two [[binary operation]]s, which are generally denoted + and · and called ''addition'' and ''multiplication'', respectively, such that:


* (''R'', +) is an [[abelian group]]
* (''R'', +) is an [[abelian group]]
* Multiplication is associative
* Multiplication is [[associative]]
* The left and right distributive laws hold:
* The left and right [[distributive law]]s hold:
** ''a''·(''b'' + ''c'') = (''a''·''b'') + (''a''·''c'')
** ''a''·(''b'' + ''c'') = (''a''·''b'') + (''a''·''c'')
** (''a'' + ''b'')·''c'' = (''a''·''c'') + (''b''·''c'')
** (''a'' + ''b'')·''c'' = (''a''·''c'') + (''b''·''c'')


In practice, the symbol · is usually omitted, and multiplication is just denoted by [[juxtaposition]]. The usual order of operations is also assumed, so that ''a'' + ''bc'' is an abbreviation for ''a'' + (''b''·''c'').
In practice, the symbol · is usually omitted, and multiplication is just denoted by [[juxtaposition]]. The usual order of operations is also assumed, so that ''a'' + ''bc'' is an abbreviation for ''a'' + (''b''·''c'').  The distributive property is specified separately for left and right multiplication to cover cases where multiplication is not commutative, such as a ring of matrices.


===Types of rings===
====Unital ring====
A ring in which there is an identity element for multiplication is called a ''unital ring'', ''unitary ring'', or simply ''ring with identity''. The identity element is generally denoted 1. Some authors, notably [[Nicholas Bourbaki|Bourbaki]], demand that their rings should have an identity element, and call rings without an identity ''pseudorings''.


===Types of Rings===
====Commutative ring====
A ring in which the multiplication operation is [[commutative law|commutative]] is called a ''commutative ring''. Such commutative rings are the basic object of study in [[commutative algebra]], in which rings are generally also assumed to have a unit.


*A ring in which there is an identity element for multiplication is called a ''unital ring'', ''unitary ring'', or simply ''ring with identity''. The identity element is generally denoted 1. Some authors, notably [[Nicholas Bourbaki|Bourbaki]], demand that their rings should have and identity element.
====Division ring====
{{Main|Division ring}}
A unital ring in which every non-zero element ''a'' has an inverse, that  is, an element ''a''<sup>&minus;1</sup> such that ''a''<sup>&minus;1</sup>''a'' = ''aa''<sup>&minus;1</sup> = 1, is called a ''division ring'' or ''skew field''.


*A ring in which the multiplication operation is [[commutative law|commutative]] is called a ''commutative ring''. Such commutative rings are the basic object of study in [[commutative algebra]], in which rings are generally also assumed to have a unit.
===Homomorphisms of rings===
A ring ''homomorphism'' is a mapping <math>\pi</math> from a ring <math>A</math> to a ring <math>B</math> respecting the ring operations. That is,
:<math>\pi(ab) = \pi(a)\pi(b)</math>
:<math>\pi(a + b) = \pi(a) + \pi(b)</math>


*A unital ring in which every element ''a'' has an inverse, that is an element ''a''<sup>&minus;1</sup> such that ''a''<sup>&minus;1</sup>''a'' = ''aa''<sup>&minus;1</sup> = 1, is called a ''division ring'' or ''skew field''.
If the rings are unital, it is often assumed that <math>\pi</math> maps the identity element of <math>A</math> to the identity element of <math>B</math>.


A homomorphism can map a larger set onto a smaller set;  for example, the ring <math>A</math> could be the integers <math>\mathbb{Z}</math> and could be mapped onto the trivial ring which contains only the single element <math>0</math>.


===Homomorphisms of Rings===
===Subrings===
A ring homomorphism is a mapping &pi; from a ring ''A'' to a ring ''B'' respecting the ring operations. That is,
If <math>A</math> is a ring, a subset <math>B</math> of <math>A</math> is called a ''subring'' if <math>B</math> is a ring under the ring operations inherited from <math>A</math>. It can be seen that this is equivalent to requiring that <math>B</math> be closed under multiplication and subtraction.
*&pi;(ab) = &pi;(a)&pi;(b)
*&pi;(a+b) = &pi;(a) + &pi;(b)


If the rings are unital, it is often assumed that &pi; maps the identity element of ''A'' to the identity element of ''B''.
If <math>A</math> is unital, some authors demand that a subring of <math>A</math> should contain the unit of <math>A</math>.


===Ideals===
A two-sided [[ideal (ring theory)|ideal]] of a ring <math>A</math> is a subring <math>I</math> such that for any element <math>a</math> in <math>A</math> and any element <math>b</math> in <math>I</math> we have that <math>ab</math> and <math>ba</math> are elements of <math>I</math>. The concept of ideal of a ring corresponds to the concept of normal subgroups of a group. Thus, we may introduce an equivalence relation on <math>A</math> by declaring that two elements of <math>A</math> are equivalent if their difference is an element of <math>I</math>. The set of equivalence classes is then denoted by <math>A/I</math> and is a ring with the induced operations.
If <math>h:A\rarr B</math> is a ring homomorphism, then the ''kernel'' of ''h'', defined as the inverse image of 0,  <math>\{x \in A:h(x) = 0\}</math>, is an ideal of <math>A</math>.  Conversely, if <math>I</math> is an ideal of <math>A</math>, then there is a natural ring homomorphism, the ''quotient homomorphism'',  from <math>A</math> to <math>A/I</math> such that <math>I</math> is the set of all elements mapped to 0 in <math>A/I</math>.


==Examples==
==Examples==
*The ''trivial ring'' {0} consists of only one element, which serves ar both additive and multiplicative identity.
*The ''trivial ring'' {0} consists of only one element, which serves as both additive and multiplicative identity.
*The [[integers]] forms a ring with addition and multiplication defined as usual. This is a [[commutative]] ring.
*The [[integers]] form a ring with addition and multiplication defined as usual. This is a [[commutative]] ring.
**The [[rational number|rational]], [[real number|real]] and [[complex number|complex]] numbers all form commutative rings.
**The [[rational number|rational]], [[real number|real]] and [[complex number|complex]] numbers each form commutative rings.
*The set of [[polynomial|polynomials]] forms a commutative ring.
*The set of [[polynomial|polynomials]] forms a commutative ring.
*The set of square <math>n\times n</math> [[matrix|matrices]] forms a ring under componentwise addition and matrix multiplication. This ring is not commutative if ''n''>1.
*The set of square <math>n\times n</math> [[matrix|matrices]] forms a ring under componentwise addition and matrix multiplication. This ring is not commutative if ''n''>1.
*The set of all [[continuous function|continuous]] real-valued [[function (mathematics)|functions]] defined on the [[interval (mathematics)|interval]] [''a'',''b''] forms a ring under pointwise addition and multiplication.
*The set of all [[continuous function|continuous]] real-valued [[function (mathematics)|functions]] defined on the [[interval (mathematics)|interval]] [''a'',''b''] forms a ring under [[pointwise operation|pointwise]] addition and multiplication.
 
 
== Basic theorems ==
 
From the axioms, one can immediately deduce that, for all elements ''a'' and ''b'' of a ring, we have
* 0''a'' = ''a''0 = 0.
* (&minus;1)''a'' = &minus;''a''.
* (&minus;''a'')''b'' = ''a''(&minus;''b'') = &minus;(''ab'').
* (''ab'')<sup>&minus;1</sup> = ''b''<sup>&minus;1</sup> ''a''<sup>&minus;1</sup> if both ''a'' and ''b'' are [[invertible]].
 
Other basic theorems
* The [[identity element]] 1 is unique.
* If the ring has at least two elements then 0 ≠ 1
* The ''[[binomial theorem]]''
::<math>(x+y)^n=\sum_{k=0}^n{n \choose k}x^ky^{n-k},</math>
 
:holds in any commutative ring.


==  Constructing new rings from given ones ==
==  Constructing new rings from given ones ==


*For every ring ''R'' we can define the '''opposite ring''' ''R''<sup>op</sup> by reversing the multiplication in ''R''. Given the multiplication · in ''R'' the multiplication ∗ in ''R''<sup>op</sup> is defined as ''b''∗''a'' := ''a''·''b''.  The "identity map" from ''R'' to ''R''<sup>op</sup> is an isomorphism if and only if ''R'' is commutative. However, even if ''R'' is not commutative, it is still possible for ''R'' and ''R''<sup>op</sup> to be isomorphic. For example, if ''R'' is the ring of ''n''&times;''n'' matrices of real numbers, then the [[transpose|transposition]] map from ''R'' to ''R''<sup>op</sup> is an isomorphism. <!-- would be nice to have an example here where the ring and its opposite are genuinely nonisomorphic -->
*For every ring <math>R</math> we can define the '''opposite ring''' <math>R^{op}</math> by reversing the multiplication in <math>R</math>. Given the multiplication <math> \cdot </math> in <math>R</math>, the multiplication <math>\star</math> in <math>R^{op}</math> is defined as <math>a \star b := b \cdot a</math>.  The "identity map" from <math>R</math> to <math>R^{op}</math>, mapping each element to itself, is an isomorphism if and only if <math>R</math> is commutative. However, even if <math>R</math> is not commutative, it is still possible for <math>R</math> and <math>R^{op}</math> to be isomorphic using a different map. For example, if <math>R</math> is the ring of <math>n \times n</math> matrices of real numbers, then the [[transpose|transposition]] map from <math>R</math> to <math>R^{op}</math>, mapping each matrix to its transpose, is an isomorphism. <!-- would be nice to have an example here where the ring and its opposite are genuinely nonisomorphic -->
* If a subset ''S'' of a ring ''R'' is closed under multiplication and subtraction and contains the multiplicative identity element, then ''S'' is called a ''[[subring]]'' of ''R''.
* The ''[[center of a ring]]'' <math>R</math> is the set of elements of <math>R</math> that commute with every element of <math>R</math>; that is, <math>c</math> is an element of the center if <math>cr = rc</math> for every <math>r \in R</math>. The center is a subring of <math>R</math>. We say that a subring <math>S</math> of <math>R</math> is central if it is a subring of the center of <math>R</math>.
* The ''[[center of a ring]]'' ''R'' is the set of elements of ''R'' that commute with every element of ''R''; that is, ''c'' lies in the center if ''cr''=''rc'' for every ''r'' in ''R''. The center is a subring of ''R''. We say that a subring ''S'' of ''R'' is central if it is a subring of the center of ''R''.
* The ''[[direct product (ring theory)|direct product]]'' of two rings ''R'' and ''S'' is the [[cartesian product]] ''R''&times;''S'' together with the operations
* The ''[[direct product (ring theory)|direct product]]'' of two rings ''R'' and ''S'' is the [[cartesian product]] ''R''&times;''S'' together with the operations
:(''r''<sub>1</sub>, ''s''<sub>1</sub>) + (''r''<sub>2</sub>, ''s''<sub>2</sub>) = (''r''<sub>1</sub>+''r''<sub>2</sub>, ''s''<sub>1</sub>+''s''<sub>2</sub>) and
:(''r''<sub>1</sub>, ''s''<sub>1</sub>) + (''r''<sub>2</sub>, ''s''<sub>2</sub>) = (''r''<sub>1</sub>+''r''<sub>2</sub>, ''s''<sub>1</sub>+''s''<sub>2</sub>) and
:(''r''<sub>1</sub>, ''s''<sub>1</sub>)(''r''<sub>2</sub>, ''s''<sub>2</sub>) = (''r''<sub>1</sub>''r''<sub>2</sub>, ''s''<sub>1</sub>''s''<sub>2</sub>).
:(''r''<sub>1</sub>, ''s''<sub>1</sub>)(''r''<sub>2</sub>, ''s''<sub>2</sub>) = (''r''<sub>1</sub>''r''<sub>2</sub>, ''s''<sub>1</sub>''s''<sub>2</sub>).
* More generally, for any index set ''J'' and collection of rings (''R''<sub>''j''</sub>)<sub>''j''ε''J''</sub>, the ''[[direct product (ring theory)|direct product]]'' and ''[[direct sum (ring theory)|direct sum]]'' exist. The direct product is the collection of "infinite-tuples"  (''r<sub>j</sub>'')<sub>''j''ε''J''</sub> with component-wise addition and multiplication. More formally, let ''U'' be the union of all of the rings ''R<sub>j</sub>''. Then the direct product of the ''R<sub>j</sub>'' over all ''j''ε''J'' is the set of all maps ''r'':''J''→''U'' with the property that ''r<sub>j</sub>''ε''R<sub>j</sub>''. Addition and multiplication of these functions is via the addition and multiplication in each individual ''R<sub>j</sub>''. Thus
:With these operations ''R''&times;''S'' is a ring.
:(''r''+''s'')<sub>''j''</sub>=''r''<sub>''j''</sub>+''s''<sub>''j''</sub> and (''rs'')<sub>''j''</sub>=''r''<sub>''j''</sub>''s''<sub>''j''</sub>.
* More generally, for any index set ''J'' and collection of rings <math>\{R_j\}_{j\in J}</math>, the ''[[direct product (ring theory)|direct product]]'' and ''[[direct sum (ring theory)|direct sum]]'' exist.  
* The direct sum of a collection of rings (''R<sub>j</sub>'')<sub>''j''ε''J''</sub> is the subring of the direct product consisting of all infinite-tuples (''r''<sub>''j''</sub>)<sub>''j''ε''J''</sub> with the property that ''r<sub>j</sub>''=0 for all but finitely many ''j''. In particular, if ''J'' is finite, then the direct sum and the direct product are isomorphic, but in general they have quite different properties.
** The direct product is the collection of "infinite-tuples"  <math>\{r_j\}_{j\in J}</math> with component-wise addition and multiplication as operations.
* Given a ring ''R'' and an [[Ideal (ring theory)|ideal]] ''I'' of ''R'', the ''[[quotient ring]]'' (or ''factor ring'') ''R''/''I'' is the set of cosets of ''I'' together with the operations
** The direct sum of a collection of rings <math>\{R_j\}_{j\in J}</math> is the subring of the direct product consisting of all infinite-tuples <math>\{r_j\}_{j\in J}</math> with the property that ''r<sub>j</sub>''=0 for all but finitely many ''j''. In particular, if ''J'' is finite, then the direct sum and the direct product are isomorphic, but in general they have quite different properties.
:(''a+I'')  + (''b+I'') = (''a''+''b'') + ''I'' and
* Since any ring is both a left and right [[module (mathematics)|module]] over itself, it is possible to construct the [[tensor product of rings|tensor product]] of ''R'' over a ring ''S'' with another ring ''T'' to get another ring, provided ''S'' is a central subring of ''R'' and ''T''.
:(''a+I'')(''b+I'') = (''ab'') + ''I''.
 
* Since any ring is both a left and right [[module (mathematics)|module]] over itself, it is possible to construct the [[tensor product of rings|tensor product]] of ''R'' over a ring ''S'' with another ring ''T'' to get another ring provided ''S'' is a central subring of ''R'' and ''T''.
==History==
The study of rings originated from the study of [[polynomial|polynomial rings]] and [[Algebraic number field|algebraic number fields]] in the second half of the nineteenth century, amongst other by [[Richard Dedekind]]. The term ''ring'' itself, however, was coined by [[David Hilbert]] in 1897.  




==See also==
==See also==


* [[Ring theory]]
* [[Glossary of ring theory]]
* [[Glossary of ring theory]]
* [[Algebra over a commutative ring]]
* [[Algebra over a commutative ring]]
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* Special types of rings:
* Special types of rings:
** [[Commutative ring]]
** [[Commutative ring]]
** [[Differential ring]]
** [[Division ring]]
** [[Division ring]]
** [[Field (mathematics)|Field]]
** [[Field (mathematics)|Field]]
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** [[Principal ideal domain]] (PID)
** [[Principal ideal domain]] (PID)
** [[Unique factorization domain]] (UFD)
** [[Unique factorization domain]] (UFD)
* Constructions of rings
** [[Group ring]]
** [[Matrix ring]]
** [[Polynomial ring]]
* Rings with added structure
** [[Differential ring]]
** [[Euclidean domain]] (ED)


==References==
==References==
Fraleigh, John B. 2003. ''A First Course in Abstract Algebra''. 7th ed. Boston: Addison-Wesley
Hilbert, David. 1897. Die Theorie der algebraische Zahlkoerper, ''ahresbericht der Deutschen Mathematiker Vereiningung'' vol. 4.
Lang, Serge. 2002. ''Algebra''. 3rd ed. New York: Springer


[[Category:Ring theory]]
<!--[[Category:Ring theory]] no such workgroup yet? -->[[Category:Suggestion Bot Tag]]
[[Category:CZ live]]

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In mathematics, a ring is an algebraic structure with two binary operations, commonly called addition and multiplication. These operations are defined so as to emulate and generalize the integers. Other common examples of rings include the ring of polynomials of one variable with real coefficients, or a ring of square matrices of a given dimension.

To qualify as a ring, addition must be commutative and each element must have an inverse under addition: for example, the additive inverse of 3 is -3. However, multiplication in general does not satisfy these properties. A ring in which multiplication is commutative and every element except the additive identity element (0) has a multiplicative inverse (reciprocal) is called a field: for example, the set of rational numbers. (The only ring in which 0 has an inverse is the trivial ring of only one element.)

A ring can have a finite or infinite number of elements. An example of a ring with a finite number of elements is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}_5} , the set of remainders when an integer is divided by 5, i.e. the set {0,1,2,3,4} with operations such as 4 + 4 = 3 because 8 has remainder 3 when divided by 5. A similar ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}_n} can be formed for other positive values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} .

Formal definition

A ring is a set R equipped with two binary operations, which are generally denoted + and · and called addition and multiplication, respectively, such that:

In practice, the symbol · is usually omitted, and multiplication is just denoted by juxtaposition. The usual order of operations is also assumed, so that a + bc is an abbreviation for a + (b·c). The distributive property is specified separately for left and right multiplication to cover cases where multiplication is not commutative, such as a ring of matrices.

Types of rings

Unital ring

A ring in which there is an identity element for multiplication is called a unital ring, unitary ring, or simply ring with identity. The identity element is generally denoted 1. Some authors, notably Bourbaki, demand that their rings should have an identity element, and call rings without an identity pseudorings.

Commutative ring

A ring in which the multiplication operation is commutative is called a commutative ring. Such commutative rings are the basic object of study in commutative algebra, in which rings are generally also assumed to have a unit.

Division ring

For more information, see: Division ring.

A unital ring in which every non-zero element a has an inverse, that is, an element a−1 such that a−1a = aa−1 = 1, is called a division ring or skew field.

Homomorphisms of rings

A ring homomorphism is a mapping Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} from a ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} to a ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} respecting the ring operations. That is,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi(ab) = \pi(a)\pi(b)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi(a + b) = \pi(a) + \pi(b)}

If the rings are unital, it is often assumed that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} maps the identity element of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} to the identity element of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} .

A homomorphism can map a larger set onto a smaller set; for example, the ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} could be the integers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}} and could be mapped onto the trivial ring which contains only the single element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} .

Subrings

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is a ring, a subset Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is called a subring if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} is a ring under the ring operations inherited from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} . It can be seen that this is equivalent to requiring that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} be closed under multiplication and subtraction.

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is unital, some authors demand that a subring of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} should contain the unit of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} .

Ideals

A two-sided ideal of a ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is a subring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} such that for any element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and any element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} we have that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ab} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ba} are elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} . The concept of ideal of a ring corresponds to the concept of normal subgroups of a group. Thus, we may introduce an equivalence relation on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} by declaring that two elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} are equivalent if their difference is an element of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} . The set of equivalence classes is then denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A/I} and is a ring with the induced operations.

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h:A\rarr B} is a ring homomorphism, then the kernel of h, defined as the inverse image of 0, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{x \in A:h(x) = 0\}} , is an ideal of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} . Conversely, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} is an ideal of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} , then there is a natural ring homomorphism, the quotient homomorphism, from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A/I} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} is the set of all elements mapped to 0 in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A/I} .

Examples

  • The trivial ring {0} consists of only one element, which serves as both additive and multiplicative identity.
  • The integers form a ring with addition and multiplication defined as usual. This is a commutative ring.
  • The set of polynomials forms a commutative ring.
  • The set of square Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\times n} matrices forms a ring under componentwise addition and matrix multiplication. This ring is not commutative if n>1.
  • The set of all continuous real-valued functions defined on the interval [a,b] forms a ring under pointwise addition and multiplication.

Constructing new rings from given ones

  • For every ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} we can define the opposite ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^{op}} by reversing the multiplication in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} . Given the multiplication Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cdot } in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} , the multiplication Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \star} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^{op}} is defined as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \star b := b \cdot a} . The "identity map" from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^{op}} , mapping each element to itself, is an isomorphism if and only if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is commutative. However, even if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is not commutative, it is still possible for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^{op}} to be isomorphic using a different map. For example, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is the ring of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \times n} matrices of real numbers, then the transposition map from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^{op}} , mapping each matrix to its transpose, is an isomorphism.
  • The center of a ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is the set of elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} that commute with every element of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} ; that is, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} is an element of the center if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle cr = rc} for every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r \in R} . The center is a subring of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} . We say that a subring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is central if it is a subring of the center of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} .
  • The direct product of two rings R and S is the cartesian product R×S together with the operations
(r1, s1) + (r2, s2) = (r1+r2, s1+s2) and
(r1, s1)(r2, s2) = (r1r2, s1s2).
With these operations R×S is a ring.
  • More generally, for any index set J and collection of rings Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{R_j\}_{j\in J}} , the direct product and direct sum exist.
    • The direct product is the collection of "infinite-tuples" with component-wise addition and multiplication as operations.
    • The direct sum of a collection of rings is the subring of the direct product consisting of all infinite-tuples with the property that rj=0 for all but finitely many j. In particular, if J is finite, then the direct sum and the direct product are isomorphic, but in general they have quite different properties.
  • Since any ring is both a left and right module over itself, it is possible to construct the tensor product of R over a ring S with another ring T to get another ring, provided S is a central subring of R and T.

History

The study of rings originated from the study of polynomial rings and algebraic number fields in the second half of the nineteenth century, amongst other by Richard Dedekind. The term ring itself, however, was coined by David Hilbert in 1897.


See also

References

Fraleigh, John B. 2003. A First Course in Abstract Algebra. 7th ed. Boston: Addison-Wesley

Hilbert, David. 1897. Die Theorie der algebraische Zahlkoerper, ahresbericht der Deutschen Mathematiker Vereiningung vol. 4.

Lang, Serge. 2002. Algebra. 3rd ed. New York: Springer

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