# subring

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.

 Title subring Canonical name Subring Date of creation 2013-03-22 12:30:19 Last modified on 2013-03-22 12:30:19 Owner yark (2760) Last modified by yark (2760) Numerical id 17 Author yark (2760) Entry type Definition Classification msc 20-00 Classification msc 16-00 Classification msc 13-00 Related topic Ideal Related topic Ring Related topic Group Related topic Subgroup Defines ideal