Context: Exercise 14, Section 3.2 of Hungerford’s Abstract Algebra.
Definition: An ideal $P$ in a ring $R$ is said to be prime if $P\neq R$ and for any ideals $A,B$ in $R$ $$AB \subseteq P \implies A\subseteq P \text{ or } B\subseteq P$$
If $P$ is an ideal in a not necessarily commutative ring $R$, then the following conditions are equivalent.
(a) P is a prime ideal.
(b) If $r,s\in R$ are such that $rRs\subseteq P$, then $r \in P$ or $s \in P$. [Hint: If $(a)$ holds and $rRs\subseteq P$, then $(RrR)(RsR)\subseteq P$, whence $RrR\subseteq P$ or $RsR\subseteq P$, say $RrR\subseteq P$. If $A= (r)$, then $A^3\subseteq RrR \subseteq P$, whence $r\in A\subseteq P$.]
(c) If $(r)$ and $(s)$ are principal ideals of $R$ such that $(r)(s)\subseteq P$, then $r\in P$ or $s\in P$.
(d) If $U$ and $V$ are right ideals in $R$ such that $UV\subseteq P$, then $U\subseteq P$ or $V\subseteq P$.
(e) If $U$ and $V$ are left ideals in $R$ such that $UV\subseteq P$, then $U\subseteq P$ or $V\subseteq P$.
There are several way to prove this exercise, for instance $(a)\Rightarrow (b)\Rightarrow (c) \Rightarrow (d) \Rightarrow (e) \Rightarrow (a)$ and $(a)\Leftrightarrow (b) \Leftrightarrow (c) \Leftrightarrow (d) \Leftrightarrow (e)$, former is the least number of implication needed to prove above exercise. But I don’t think it is the most easiest way to do it.
I want to prove this exercise by showing following implications (which I think is the easiest approach): $(a) \Leftrightarrow (c)$ and $(a) \Rightarrow (b) \Rightarrow (d) \Rightarrow (a)$. I am not considering $(e)$, because $(d)$ can be replaced by $(e)$ with slight modification.
Proof of $(a) \Leftrightarrow (c)$is fairly easy. Proof of $(b)\Rightarrow (d)$ is here (below $\neg a\implies \neg b$). Proof of $(d)\Rightarrow (a)$ is trivial. So only thing left to show is that $(a)\Rightarrow (b)$. Though given hint is quite straightforward. I am having hard time showing $RrR$ is an ideal, to be specific $RrR$ is closed under addition. Let $arb, crd\in RrR$. How to show $arb+crd=xry$ for some $x,y\in R$? I tryed using $x=a+c$ and $y=b+d$. But this don’t work. $RrR$ is closed under addition is also used in filling details of $A^3\subseteq RrR$ (see here).
Hint： Can you prove that for left ideals $A_1,A_2,\dots ,A_n$ , $A_1A_2\dots A_n$ is still a left ideal?