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A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082400/r0824001.png" /> on which two binary algebraic operations are defined: addition and multiplication, the set being an [[Abelian group|Abelian group]] (the additive group of the ring) with respect to addition, and the multiplication is related to the addition by the distributive laws:
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A set $R$ on which two binary algebraic operations are defined: addition and multiplication, the set being an [[Abelian group|Abelian group]] (the additive group of the ring) with respect to addition, and the multiplication is related to the addition by the distributive laws:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082400/r0824002.png" /></td> </tr></table>
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$$a(b+c)=ab+ac,\quad(b+c)a=ba+ca,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082400/r0824003.png" />. In general no restriction is imposed on multiplication, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082400/r0824004.png" /> is a [[Groupoid|groupoid]] (called the multiplicative groupoid of the ring) with respect to multiplication.
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where $a,b,c\in R$. In general no restriction is imposed on multiplication, that is, $R$ is a [[Groupoid|groupoid]] (called the multiplicative groupoid of the ring) with respect to multiplication.
  
A non-empty subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082400/r0824005.png" /> is called a subring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082400/r0824006.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082400/r0824007.png" /> itself is a ring with respect to the operations defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082400/r0824008.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082400/r0824009.png" /> must be a subgroup of the additive group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082400/r08240010.png" /> and a subgroupoid of the multiplicative groupoid of this ring. Clearly, the ring itself and the zero subring consisting of just the zero element are subrings of a given ring. The (set-theoretic) intersection of subrings of a ring is a subring. The join of a family of subrings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082400/r08240011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082400/r08240012.png" />, of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082400/r08240013.png" /> is the intersection of all subrings that contain all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082400/r08240014.png" />. The set of all subrings of a given ring is a [[Lattice|lattice]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082400/r08240015.png" />, with respect to the operations of intersection and join of subrings. The set of ideals (cf. [[Ideal|Ideal]]) of this ring forms a sublattice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082400/r08240016.png" />.
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A non-empty subset $A\subset R$ is called a subring of $R$ if $A$ itself is a ring with respect to the operations defined on $R$, that is, $A$ must be a subgroup of the additive group of $R$ and a subgroupoid of the multiplicative groupoid of this ring. Clearly, the ring itself and the zero subring consisting of just the zero element are subrings of a given ring. The (set-theoretic) intersection of subrings of a ring is a subring. The join of a family of subrings $A_\alpha$, $\alpha\in I$, of a ring $R$ is the intersection of all subrings that contain all $A_\alpha$. The set of all subrings of a given ring is a [[Lattice|lattice]], $S(R)$, with respect to the operations of intersection and join of subrings. The set of ideals (cf. [[Ideal|Ideal]]) of this ring forms a sublattice of $S(R)$.
  
 
Concerning the various directions in the theory of rings, see [[Rings and algebras|Rings and algebras]]; [[Associative rings and algebras|Associative rings and algebras]]; [[Non-associative rings and algebras|Non-associative rings and algebras]].
 
Concerning the various directions in the theory of rings, see [[Rings and algebras|Rings and algebras]]; [[Associative rings and algebras|Associative rings and algebras]]; [[Non-associative rings and algebras|Non-associative rings and algebras]].
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====Comments====
 
====Comments====
In many contexts it is tacitly assumed that the ring contains a unit element, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082400/r08240017.png" />, and subrings are taken to be subrings with the same unit. In this case the set of ideals is not a sublattice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082400/r08240018.png" />.
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In many contexts it is tacitly assumed that the ring contains a unit element, denoted by $1$, and subrings are taken to be subrings with the same unit. In this case the set of ideals is not a sublattice of $S(R)$.

Revision as of 14:15, 30 July 2014

A set $R$ on which two binary algebraic operations are defined: addition and multiplication, the set being an Abelian group (the additive group of the ring) with respect to addition, and the multiplication is related to the addition by the distributive laws:

$$a(b+c)=ab+ac,\quad(b+c)a=ba+ca,$$

where $a,b,c\in R$. In general no restriction is imposed on multiplication, that is, $R$ is a groupoid (called the multiplicative groupoid of the ring) with respect to multiplication.

A non-empty subset $A\subset R$ is called a subring of $R$ if $A$ itself is a ring with respect to the operations defined on $R$, that is, $A$ must be a subgroup of the additive group of $R$ and a subgroupoid of the multiplicative groupoid of this ring. Clearly, the ring itself and the zero subring consisting of just the zero element are subrings of a given ring. The (set-theoretic) intersection of subrings of a ring is a subring. The join of a family of subrings $A_\alpha$, $\alpha\in I$, of a ring $R$ is the intersection of all subrings that contain all $A_\alpha$. The set of all subrings of a given ring is a lattice, $S(R)$, with respect to the operations of intersection and join of subrings. The set of ideals (cf. Ideal) of this ring forms a sublattice of $S(R)$.

Concerning the various directions in the theory of rings, see Rings and algebras; Associative rings and algebras; Non-associative rings and algebras.


Comments

In many contexts it is tacitly assumed that the ring contains a unit element, denoted by $1$, and subrings are taken to be subrings with the same unit. In this case the set of ideals is not a sublattice of $S(R)$.

How to Cite This Entry:
Ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ring&oldid=32563
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article