Normal and tangential flatness.
First, we give a complete description of the indecomposable prime modules over a Dedekind domain. Second, if is the pullback, in the sense of [9], of two local Dedekind domains then we classify indecomposable prime -modules and establish a connection between the prime modules and the pure-injective modules (also representable modules) over such rings.
In this paper, we define Gorenstein injective rings, Gorenstein injective modules and their envelopes. The main topic of this paper is to show that if is a Gorenstein integral domain and is a left -module, then the torsion submodule of Gorenstein injective envelope of is also Gorenstein injective. We can also show that if is a torsion -module of a Gorenstein injective integral domain , then the Gorenstein injective envelope of is torsion.
All rings considered are commutative with unit. A ring R is SISI (in Vámos' terminology) if every subdirectly irreducible factor ring R/I is self-injective. SISI rings include Noetherian rings, Morita rings and almost maximal valuation rings ([V1]). In [F3] we raised the question of whether a polynomial ring R[x] over a SISI ring R is again SISI. In this paper we show this is not the case.
The aim of this paper is to discuss the flat covers of injective modules over a Noetherian ring. Let R be a commutative Noetherian ring and let E be an injective R-module. We prove that the flat cover of E is isomorphic to . As a consequence, we give an answer to Xu’s question [10, 4.4.9]: for a prime ideal p, when does appear in the flat cover of E(R/m̲)?
In this paper the concept of the second submodule (the dual notion of prime submodule) is introduced.