Generalized ideal transforms

Communications in Algebra, 2012

In this paper we study when the dual of a t-ideal in a P V M D is a ring? and we treat the question when it coincides with its endomorphism ring. We also study particular classes of overrings of PVMDs. Specially, we investigate the Nagata transform and the endomorphism ring of ideals in PVMDs in an attempt to establish analogues for well-known results on overrings of Prüfer domains.

Promoter: Professor N. J. Groenewald 0.1. ABSTRACT i 0.1 Abstract This thesis is aimed at generalizing notions of rings to modules. In particular, notions of completely prime ideals, s-prime ideals, 2-primal rings and nilpotency of elements of rings are respectively generalized to completely prime submodules and classical completely prime submodules, s-prime submodules, 2-primal modules and nilpotency of elements of modules. Properties and radicals that arise from each of these notions are studied. ii 0.2 Acknowledgement Firstly, I express gratitude to my promoter Prof. N. J. Groenewald. His invaluable assistance, guidance and advice which was more than just academic led me this far. I am indebted to DAAD, NRF and NMMU for the financial support. I thank Prof. Straeuli, Prof. Booth, Ms Esterhuizen and all staff of the Department of Mathematics and Applied Mathematics of NMMU for being hospitable and for providing an environment conducive for learning -baie dankie! I owe gratitude to Prof. G. K.

Journal of Algebra, 1987

Promoter: Professor N. J. Groenewald 0.1. ABSTRACT i 0.1 Abstract This thesis is aimed at generalizing notions of rings to modules. In particular, notions of completely prime ideals, s-prime ideals, 2-primal rings and nilpotency of elements of rings are respectively generalized to completely prime submodules and classical completely prime submodules, s-prime submodules, 2-primal modules and nilpotency of elements of modules. Properties and radicals that arise from each of these notions are studied. ii 0.2 Acknowledgement Firstly, I express gratitude to my promoter Prof. N. J. Groenewald. His invaluable assistance, guidance and advice which was more than just academic led me this far. I am indebted to DAAD, NRF and NMMU for the financial support. I thank Prof. Straeuli, Prof. Booth, Ms Esterhuizen and all staff of the Department of Mathematics and Applied Mathematics of NMMU for being hospitable and for providing an environment conducive for learning -baie dankie! I owe gratitude to Prof. G. K.

We study pairs of finitely generated modules over a principal ideal domain and their corresponding matrix representations. We introduce equivalence relations for such pairs and determine invariants and canonical forms.

2014

2 Modules over a Principal Ideal Domain 6 2.1 Review of principal ideal domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Structure of finitely generated modules over a PID: Cyclic Decomposition . . . . . . . . . . . 6 2.3 Equivalence of matrices over a PID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Presentation Matrices: generators and relations . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Journal of Pure and Applied Algebra, 2007

KrulΓs principal ideal theorm [Krull] states that q elements in the maximal ideal of a local noetherian ring generate an ideal whose minimal components are all of height at most q. Writing R for the ring, we may consider the q elements, x 19 , x q say, as coordinates of an element xeR q .

2001

Using symmetric algebras we simplify (and slightly strengthen) the Bruns-Eisenbud-Evans "generalized principal ideal theorem" on the height of order ideals of nonminimal generators in a module. We also obtain a simple proof and an extension of a result by Kwieciński, which estimates the height of certain Fitting ideals of modules having an equidimensional symmetric algebra.

Journal of Mathematical Sciences, 1999

Proceedings of the Japan Academy, 1971