Given a ring , a subset is called a subring of if it inherits the ring structure from . That is, must contain both the and (additive and multiplicative identities) of and be closed under the ring operations of multiplication, addition and additive inverse-taking.
Consider the ring of ordered pairs of integers with coordinatewise operations, i.e. and . Then the diagonal ring is a subring of : it contains the additive identity , the multiplicative identity and is closed under multiplication and addition.
The notion of a subring is slightly more subtle than that of a subgroup. Suppose that is a commutative ring with an idempotent element other than and , i.e. is a solution to the equation . Consider the principle ideal . As an ideal, this set is closed under addition and multiplication and contains the additive identity of . Moreover, this ideal is a ring with multiplicative identity : for every , so for every . However, it is not a subring of because it does not contain the multiplicative identity of . (Otherwise and there is some such that , so but also , and we assumed , a contradiction.)
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