Subring
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.
Examples
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.
Non-examples
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.)
See also
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