On finitely generated projective modules and multiplication modules

This paper is based on the M. Sc. thesis written by the third author under the supervision of the first two authors. It was submitted to the University of Baghdad in January 1987.     

Comultiplication submodules of multiplication modules

Abstract

All rings are commutative with identity, and all modules are unital. The purpose of this work is to investigate comultiplication submodules of multiplication modules. Various properties of this class of submodules are considered. Sufficient conditions for the sum and intersection of a finite collection of comultiplication submodules to be a comultiplication submodule are also given.     

On finiteness of multiplication modules

Our main aim in this note, is a further generalization of a result due to D. D. Anderson, i.e., it is shown that if R is a commutative ring, and M a multiplication R-module, such that every prime ideal minimal over Ann (M) is finitely generated, then M contains only a finite number of minimal prime submodules. This immediately yields that if P is a projective ideal of R, such that every prime ideal minimal over Ann (P) is finitely generated, then P is finitely generated. Furthermore, it is established that if M is a multiplication R-module in which every minimal prime submodule is finitely generated, then R contains only a finite number of prime ideals minimal over Ann (M).      

On finitely generated multiplication modules

We shall prove that if M is a finitely generated multiplication module and Ann(M) is a finitely generated ideal of R, then there exists a distributive lattice M such that Spec(M) with Zariski topology is homeomorphic to Spec(M) to Stone topology. Finally we shall give a characterization of finitely generated multiplication R-modules M such that Ann(M) is a finitely generated ideal of R.     

On multiplication and comultiplication modules

This article deals with some results concerning multiplication and comultiplication modules over a commutative ring.     

Pseudo-absorbing multiplication modules over a pullback ring

In this paper, we introduce the notion of pseudo-absorbing multiplication modules. The purpose of this paper is to outline an approach to the classification of indecomposable pseudo-absorbing multiplication modules with finite-dimensional top over certain kinds of pullback rings and establish a connection between the pseudo-absorbing multiplication modules and the pure-injective modules over such rings.     

Reduced rings whose finitely generated modules decompose

This paper proves Hua's Theorem and, from this, the Fundamental Theorem of Projective Geometry for the projective line over a commutative ring with “many units”.     

Minimal models for Drinfeld modules of rank 2 with complex multiplication

We show that some Drinfeld modules of rank 2 on Bbb F_q[T]{Bbb F}_q[T] with complex multiplication have global minimal models.     

Crossed modules and doi-hopf modules

We prove that crossed modules (or Yetter-Drinfel’d modules) are special cases of Doi’s unified Hopf modules. The category of crossedH-modules is therefore a Grothendieck category (if we work over a field), and the Drinfel’d double appears as a type of generalized smash product. The second and the third author both thank the University of Brussels for its warm hospitality during their visit there. This author was supported partially by the CNCSU nr. 221 and by the Romanian academy, grant nr. 3721.