# Subrings

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

**(i) **

**(ii)**

__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 .