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