On almost maximal right ideals

A concept of “a prime ideal” in a commutative ring is extended to a general ring such that it properly includes the class of maximal one sided ideals. Such a right (or left) ideal is called almost maximal. The main theorems in the present paper are as follows: (1) If $R$ is a ring with 1 then a right ideal $I$ is almost maximal if and only if ${\operatorname {Hom} _R}({[R/I]_0},{[R/I]_0})$ is a division ring where ${[R/I]_0}$ is the quasi-injective hull of $R/I$, and for any nonzero submodule $N$ of $R/I$ there is a nonzero endomorphism $f$ of $R/I$ such that $f(R/I) \subset N$. (2) If $R$ is a ring with $1$ then $R$ is a right noetherian ring and every almost maximal right ideal is maximal if and only if $R$ is a right artinian ring.