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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On almost maximal right ideals


Author: Kwangil Koh
Journal: Proc. Amer. Math. Soc. 25 (1970), 266-272
MSC: Primary 16.20
DOI: https://doi.org/10.1090/S0002-9939-1970-0265393-8
MathSciNet review: 0265393
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Abstract: 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.


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DOI: https://doi.org/10.1090/S0002-9939-1970-0265393-8
Keywords: Normalizer, quasi-injective hull, noetherian ring, artinian ring, Goldie ring, strongly regular ring
Article copyright: © Copyright 1970 American Mathematical Society