Let be a ring. A non–empty subset of the set is said to be a subring of if is closed under addition and multiplication in and itself is a ring for those operations.
If is any ring, then and are always subrings of . These are said to be improper subrings. The subrings of other than these two, if any, are said to be proper subrings of .
Theorem: The necessary and sufficient conditions for a non-empty subset of a ring to be a subring of are
Proof: To prove that the conditions are necessary let us suppose that is a subring of . Obviously is a group with respect to addition, therefore .
Since is closed under addition,
Also is closed with respect to multiplication,
Now to prove that the conditions are sufficient, suppose is a non-empty subset of for which the conditions (i) and (ii) are satisfied.
From condition (i)
Hence additive identity is in . Now
i.e. each element of possesses additive inverse.
Let then and then from condition (i)
Thus is closed under addition, and being a subset of , associative and commutative laws of multiplication over addition holds in . Thus is a subring of .