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subring (Definition)

Let $(R,+,*)$ a ring. A subring is a subset $S$ of $R$ with the operations $+$ and $*$ of $R$ restricted to $S$ and such that $S$ is a ring by itself.

Notice that the restricted operations inherit the associative and distributive properties of $+$ and $*$, as well as commutativity of $+$. So for $(S,+,*)$ to be a ring by itself, we need that $(S,+)$ be a subgroup of $(R,+)$ and that $(S,*)$ be closed. The subgroup condition is equivalent to $S$ being non-empty and having the property that $x-y\in S$ for all $x,y\in S$.

A subring $S$ is called a left ideal if for all $s\in S$ and all $r\in R$ we have $r*s\in S$. Right ideals are defined similarly, with $s*r$ instead of $r*s$. If $S$ is both a left ideal and a right ideal, then it is called a two-sided ideal. If $R$ is commutative, then all three definitions coincide. In ring theory, ideals are far more important than subrings, as they play a role analogous to normal subgroups in group theory.

Example:

Consider the ring $(\mathbb{Z},+,\cdot$). Then $(2\mathbb{Z},+,\cdot)$ is a subring, since the difference and product of two even numbers is again an even number.



"subring" is owned by yark. [ full author list (2) | owner history (2) ]
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See Also: ideal, ring, group, subgroup

Also defines:  ideal

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generated subring (Definition) by polarbear
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Cross-references: even numbers, product, difference, group, normal subgroups, definitions, two-sided ideal, right ideals, left ideal, subgroup, commutativity, distributive properties, associative, operations, subset, ring
There are 126 references to this entry.

This is version 14 of subring, born on 2002-03-02, modified 2005-12-16.
Object id is 2738, canonical name is Subring.
Accessed 9428 times total.

Classification:
AMS MSC: 13-00 (Commutative rings and algebras :: General reference works )
 16-00 (Associative rings and algebras :: General reference works )
 20-00 (Group theory and generalizations :: General reference works )
Pending Errata and Addenda
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Subring criterion by pahio on 2004-06-16 17:44:11

The subset S of Z[X] which consists of polynomials with positive constant term is closed under addition and multiplication, but S is not subring of Z[X].

The sufficient (and necessary) subring criterion is that the non-empty S is closed under subtraction and multiplication.
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