In abstract algebra
, the subring test
is a theorem
that states that for any ring
, a nonempty subset
of that ring is a subring
if it is closed
under multiplication and subtraction. Note that here that the terms ring
are used without requiring a multiplicative identity element.
More formally, let be a ring, and let be a nonempty a subset of . If for all one has and for all one has then is a subring of .
If rings are required to have unity, then it must also be assumed that the multiplicative identity is in the subset.
is nonempty and closed under subtraction, by the subgroup test
it follows that
is a group
under addition. Hence,
is closed under addition, addition is associative,
has an additive identity, and every element in
has an additive inverse.
Since the operations of are the same as those of it immediately follows that addition is commutative, multiplication is associative, multiplication is left distributive over addition, and multiplication is right distributive over addition.
Thus, is a subring of .