Unity in Subrings, Zero Divisors

Hello!

I have two questions!

(1)

if $\displaystyle M$ is a subring of $\displaystyle N$, where $\displaystyle N$ has unity (let's call it $\displaystyle 1_N$), and say $\displaystyle 1_M$ is the unity of $\displaystyle M$. Now, I have shown that it is NOT necessarily true that $\displaystyle 1_M = 1_N$

Then if $\displaystyle M$ and $\displaystyle N$ are both fields (every element is a unit) then I'm trying to show that it is necessarily true that $\displaystyle 1_M = 1_N$.

I am thinking you just take any $\displaystyle m \in M$, then $\displaystyle m*m^{-1} = 1_M$, but since $\displaystyle m \in N$ as well, $\displaystyle m*m^{-1} = 1_N$. Is that legitimate reasoning? I'm not confident with my answer for some reason.

(2) Let $\displaystyle m,n \in R$ for a commutative ring $\displaystyle R$.

Suppose that $\displaystyle mn$ is a zero divisor in $\displaystyle R$. Prove that either $\displaystyle m$ or $\displaystyle n$ is a zero divisor.

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

If $\displaystyle mn \neq 0$ is a z.d. then $\displaystyle \exists x \in R$ such that $\displaystyle x \neq 0 $ and $\displaystyle mn(x) = 0$

Then, just by associativity, $\displaystyle m(nx) = 0$ so, well, **can we assume that** $\displaystyle nx \neq 0$ and $\displaystyle m \neq 0$? Cause we'd be done - $\displaystyle m$ is a z.d. ! By commutativity, just switch $\displaystyle m$ and $\displaystyle n$ and then we're done.

Any help appreciated! Thanks!!