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On Wikipedia it says that an ideal $I \neq R$ in a non-commutative ring $R$ is prime if whenever two ideals $A,B$ satisfy $AB \subseteq I$ then either $A \subseteq I$ or $B \subseteq I$. It also mentions that there are 4 other equivalent definitions.

I was wondering if there are one-sided ideals $I$ satisfying any one of these 5 conditions, but are not two-sided ideals themselves (and therefore are not considered prime)?

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Sure. Let $N$ be any proper right ideal of $R=M_n(F)$ for a field $F$.

Then $N$ satisfies the very first definition: if $A,B$ are ideals of $R$ such that $AB\subseteq N$, then $A$ or $B$ is contained in $N$. For $AB\subseteq N$, one of $A$ or $B$ is zero. (There are only two ideals in $R$!)

There are, however, interesting notions of primeness for one-sided ideals. An especially interesting one is the major one in this paper.

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