On the trivialization of line bundles over schemes
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 class of pure submodules () and torsion-free images () of finite direct sums of submodules of the quotient field of an integral domain were first investigated by M. C. R. Butler for the ring of integers (1965). In this case and short exact sequences of such modules are both prebalanced and precobalanced. This does not hold for integral domains in general. In this paper the notion of precobalanced sequences of modules is further investigated. It is shown that as in the case for abelian groups...
It is a well-known fact that modules over a commutative ring in general cannot be classified, and it is also well-known that we have to impose severe restrictions on either the ring or on the class of modules to solve this problem. One of the restrictions on the modules comes from freeness assumptions which have been intensively studied in recent decades. Two interesting, distinct but typical examples are the papers by Blass [1] and Eklof [8], both jointly with Shelah. In the first case the authors...