An injective-envelope-based characterization of distributive modules over commutative Noetherian rings

Abstract

Let R be a commutative Noetherian ring and M be an R-module. The R-module M is called distributive if for every submodules S, T and U of M, the equality $S\cap (T+U)=S\cap T+S\cap U$ holds true. In this paper, we give a necessary and sufficient condition for M to be distributive based on injective envelopes. The proof uses Matlis’ results on injective modules.

KEYWORDS:

• Dedekind domain
• discrete valuation ring
• distributive module
• injective envelope
• Matlis duality
• orthogonality
• Prüfer domain
• splitting relative to the direct sum decomposition
• uniserial module

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 13E05
• 13C11
• Secondary: 13F10
• 13H15