Property of Zero element in factor ring

Question

I'm trying to learn proof for "Let R be a commutative ring with unity and let A be an ideal of R.Then R/A is an integral domain if and only if A is prime". In first part of proof in text book I see this statement:

MY PROBLEM IS: 1:Why ab is the zero element of the ring R/A? 2:I'm confused with properties of "general" R/A ring, what is similar and not similar to the factor groups? IMPORTANT NOTE: I have read lots of similar related topics but there wasn't any helpful or explanation about my questions.

Answer 1

1. As a set, $R/A = \{a + A \big| \ a\in R \}$, the set of equivalence classes modulo $A$. Now, the zero of the ring $R/A$ is the equivalence class of $0_R$ ie $0_R + A$, but $0_R + A = A$. That's why the author of this proof says that "$A$ is the zero element of the ring $R/A$.

2. Quotient groups and quotient rings work alike. The difference is that while you quotient groups by normal subgroups, you don't quotient rings by rings but by ideals instead.

Answer 2

Because $ab \in A$ we have $ab + A = 0 + A$ in $R/A$, and $0 + A$ is the zero element.

The ring $R/A$ is exactly a factor group. The ring $R$ under addition is an abelian group called the additive group of $R$. The ideal $A$ is a subgroup of the additive group of $R$. Since the additive group is abelian all subgroups are normal, so the factor group $R/A$ exists. This factor group is the additive group of the ring $R/A$, in fact the construction of the ring $R/A$ is done by defining a multiplication operation on this factor group.