L2-PRIME AND DIMENSIONAL MODULES

Abstract

We introduce a map κ that generalizes Krull and Noetherian dimensions. If MR finitely generates all fully invariant submodules and has acc on them, there are only a finite number of minimal L2-prime submodules Pi(1 ≤ i ≤ n) and when defined, κ(M) = κ(M/Pj ) for some j. Here, each M/Pi is a prime R-module, and in particular, M has finite length if every irreducible prime submodule of M is maximal. Quasi-projective L2-prime Rmodule with non-zero socle are investigated and some applications are then given when κ(M) means the Krull dimension or the injective dimension.