# Intersection of Subrings

__Theorem__**:**

The intersection of two subrings is a subring.

__Proof__**:**

Let and be two subrings of ring .

Since and at least . Therefore is non-empty.

Let , then

and

and

and

But and are subrings of , therefore

and

and

and

Consequently, and .

Hence, is a subring of .