ABSTRACT
Bruce Olberding Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003-8001, USA olberdin@emmy.nmsu.edu
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 12.2 The Structure of Q-irreducible Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 12.3 Completely Q-Irreducible and m-Canonical Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 12.4 Q-irreducibility and Injective Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 12.5 Irredundant Decompositions and Semi-Artinian Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 12.6 Pru¨fer Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 12.7 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 12.8 Appendix: Corrections to [17] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Abstract Let R be an integral domain and let Q denote the quotient field of R. We investigate the structure of R-submodules of Q that are Q-irreducible, or completely Q-irreducible. One of our goals is to describe the integral domains that admit a completely Q-irreducible ideal, or a nonzero Q-irreducible ideal. If R has a nonzero finitely generated Q-irreducible ideal, then R is quasilocal. If R is integrally closed and admits a nonzero principal Q-irreducible ideal, then R is a valuation domain. If R has an m-canonical ideal and admits a completely Q-irreducible ideal, then R is quasilocal and all the completely Q-irreducible ideals of R are isomorphic. We consider the condition that every nonzero ideal of R is an irredundant intersection of completely Q-irreducible submodules of Q and present eleven conditions that are equivalent to this. We classify the domains for which every nonzero ideal can be represented uniquely as an irredundant intersection of completely Q-irreducible submodules of Q. The domains with this property are the Pru¨fer domains that are almost semi-artinian, that is, every proper homomorphic image has a nonzero socle. We characterize the Pru¨fer and Noetherian domains that possess a completely Q-irreducible ideal or a nonzero Q-irreducible ideal. Subject classifications: Primary 13A15, 13F05. Keywords: irreducible ideal, completely irreducible ideal, injective module, Pru¨fer domain, m-
12.1 Introduction This article continues a study of commutative ideal theory in rings without finiteness conditions begun in [15], [16], [17] and [26]. In [15] and [16] we examine irreducible and completely irreducible ideals of commutative rings. In the present article we investigate stronger versions of these two notions of irreducibility for ideals of integral domains. In particular, we consider irreducibility of an ideal of an integral domain when it is viewed as a submodule of the quotient field of the domain.
