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