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# 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.

## Mathematics Subject Classification

20-00*no label found*16-00

*no label found*13-00

*no label found*

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## Corrections

subring by pahio ✓

not ideal by pahio ✓

The description of a left ideal of a ring. by ponienchen ✓

Slip of the pen by pahio ✓

## Comments

## Subring criterion

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.

## Re: Subring criterion

the proper way is not to post a message but to file a correction, no point in doing both

f

G -----> H G

p \ /_ ----- ~ f(G)

\ / f ker f

G/ker f