Commutative Ideal Theory without Finiteness Conditions: Irreducibility in the Quotient Field

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 Prüfer domains that are almost semiartinian, that is, every proper homomorphic image has a nonzero socle. We characterize the Prüfer or Noetherian domains that possess a completely Qirreducible ideal or a nonzero Q-irreducible ideal.