I am reading Atiyah and MacDonald's 'Introduction to Commutative Algebra' and wished to check a point in one of their proofs. They state that if $A$ is a non-zero ring the following are equivalent:
(i.) $A$ is a field
(ii.) The only ideals in $A$ are $0$ and $(1)$
(iii.) Every homomorphism of $A$ to a non-zero ring $B$ is injective.
To prove that (iii.) implies (i.) they say 'Let $x$ be an element of $A$ which is not a unit. Then $(x)$ does not equal $(1)$, hence $B=A/(x)$ is not the zero ring'. To fill in the gaps here, is that because the principal ideal generated by 1 would return the original ring $A$ because you are just multiplying $1$ through by every element of the ring one-by-one, and then that gives you $B=A/A$, which is the zero ring? If anyone wants to fill in the gaps a bit more explicitly I would appreciate it.
Apologies about the basic question, as I am new to elementary abstract algebra.