On almost maximal right ideals

Abstract

A concept of prime 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 is almost maximal if and only if Hom0R([R/I]o, [R/11]0) is a division ring where [R/II]o 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) CN. (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. 1. If R is a ring and is a right ideal of R, let N(I) = { xCER |xI CI } and N*(I) = {xEN(I) xyCI if and only if yEEI}. N(I) is called the normalizer of I in R and it is the largest subring of R in which is contained as an ideal. (Refer [4, p. 25].) Let R\I denote the (set) complement of in R. We define a right ideal which is not R to be almost maximal if and only if (Al) for any a, bER\I there are ri, r2ER and cEN*(I) such that ar1 br2 -c mod I, (A2) if aER\I, then either aEN(I) or ar=ai mod for some rER\I and iEI. If R is a commutative ring then (A2) is true always for any ideal and the condition (Al) holds true if and only if is a prime ideal. The purpose of this paper is to prove the following theorems: (1) If R is a ring with 1 then a right ideal is almost maximal if and only if (i) HomR( [R/I]o, [R/I]o) is a division ring, where [R/I]o is the quasi-injective hull of R/I, (ii) if N is a nonzero submodule of R/I then there is a nonzero endomorphism f in HomR (R/I, R/I) such that f (R/I) CN. (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. (2) is a generalization of Theorem 2 of [7, p. 205]. Received by the editors July 25, 1969. AMS Subject Classifications. Primary 1620, 1625, 1640.