Unity in Subrings, Zero Divisors

Hello!

I have two questions!

(1)

if is a subring of , where has unity (let's call it ), and say is the unity of . Now, I have shown that it is NOT necessarily true that

Then if and are both fields (every element is a unit) then I'm trying to show that it is necessarily true that .

I am thinking you just take any , then , but since as well, . Is that legitimate reasoning? I'm not confident with my answer for some reason.

(2) Let for a commutative ring .

Suppose that is a zero divisor in . Prove that either or is a zero divisor.

So, Well, first what I did was use the defn of zero divisor (z.d.)

If is a z.d. then such that and

Then, just by associativity, so, well, **can we assume that** and ? Cause we'd be done - is a z.d. ! By commutativity, just switch and and then we're done.

Any help appreciated! Thanks!!