## How to prove the intersection of two subrings is a subring?

Let S1 and S2 be two subrings of a ring R. Then S1∩S2 is not empty since at least 0⋿ S1∩S2.

Now in order to prove that S1∩S2 is subring, it is sufficient to prove that

1. a⋿ S1∩S2,b⋿ S1∩S2

a-b⋿ S1∩S2

2. a⋿ S1∩S2,b⋿ S1∩S2

ab⋿ S1∩S2

We have

a⋿ S1∩S2

a⋿ S1,a⋿ S2

b⋿ S1∩S2

b⋿ S1,b⋿ S2

### Now S1 and S2 are both subrings.

such that

a⋿ S1,b⋿ S1

a-b⋿ S1 and ab⋿ S1

and a⋿ S2,b⋿ S2

ab⋿ S2.

Now a-b ⋿ S1,a-b⋿ S2

a-b⋿ S1∩S2

ab⋿ S1,ab⋿ S2

ab⋿ S1∩S2.

Thus

a⋿ S1∩S2,

b⋿ S1∩S2

a-b⋿ S1∩S2.and

ab⋿ S1∩S2.

so that

S1∩S2 is a subring of R.