0. Ring ideals

Let $R$ be a commutative ring with 1. As you know,

(1) an ideal of $R$ is a nonempty subset $A$ of $R$ such that $a _ 1 + a _ 2 \in A$ for all $a _ 1, a _ 2 \in A$, and $ra \in A$ for all $a \in A, r \in R$;

(2) an ideal $A$ is principal if $A = (a) = \{ra \vert r \in R\}$ for some $a \in A$;

(3) an ideal $A$ is proper if $A < R$.

(4) a ideal $A$ is prime if $A$ is proper and $xy \in A$ implies $x \in A$ or $y \in A$;

(5) an ideal $A$ is maximal (meaning maximal proper) if $A$ is proper and there is no ideal $I$ with $A < I < R$.