I am having trouble with another homework problem (Hungerford, Algebra, Problem 12, Chapter III, Section 2).
Let $R$ be a ring without identity and with no zero divisors. Let $S$ be the ring whose additive group is $R \times \mathbb{Z}$ with multiplication given by $(r_1,n_1)(r_2,n_2)=(r_1r_2+n_2r_1+n_1r_2,n_1n_2).$ Let $A=\{(r,n) \in S \mid rx+nx=0 \text{ for all } x \in R\}$.
(a) $A$ is an ideal in $S$.
(b) $S/A$ has an identity and contains a subring isomorphic to $R$.
(c) $S/A$ has no zero divisors.
I've done (a) and (b), but haven't been able to crack (c). Here's what I have so far:
(c) Suppose $((r, n) + A)((s, m) + A) = (0, 0) + A$. Then $(r, n)(s, m) \in A$, so $(rs + mr + ns)x + (nm)x = 0$ for all $x \in R$. But $(rs + mr + ns)x + (nm)x = rsx + mrx + nsx + nmx = r(sx + mx) + n(sx + mx)$.
Now here we have an expression involving $r(something) + n(something)$ and $s(something) + m(something)$, and I want to say that if it's zero for all $x$ then $rx + nx = 0$ for all $x$ or $sx + mx = 0$ for all $x$, but all I seem to be able to say is that $rx + nx = 0$ for all $x = sx' + mx'$, which is not (or not obviously) what I want.
I appreciate any help!