A Prime Ideal Principle for Two-Sided Ideals
Manuel Reyes2016, Communications in Algebra
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Abstract
Many classical ring-theoretic results state that an ideal that is maximal with respect to satisfying a special property must be prime. We present a "Prime Ideal Principle" that gives a uniform method of proving such facts, generalizing the Prime Ideal Principle for commutative rings due to T. Y. Lam and the author. Old and new "maximal implies prime" results are presented, with results touching on annihilator ideals, polynomial identity rings, the Artin-Rees property, Dedekind-finite rings, principal ideals generated by normal elements, strongly noetherian algebras, and just infinite algebras. Conventions. Throughout this paper, all rings, modules, and ring homomorphisms are assumed to be unital. Let R be a ring. We write I R to denote that I is an ideal of R. If X is a subset of R, we write (X) for the ideal of R generated by X; in case X = {x} is a singleton we simply write (x) = ({x}). Furthermore, if J R and a ∈ R, then we denote (I, J) = (I ∪ J) = I + J and (I, a) = (I ∪ {a}) = I + (a). The notation M R and R M respectively indicate that M is a right or left R-module.
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