We study the relation of the ideal class group of a Dedekind domain A to that of As, where S is a multiplicatively closed subset of A. We construct examples of (a) a Dedekind domain with no principal prime ideal and (b) a Dedekind domain which is not the integral closure of a principal ideal domain. We also obtain some qualitative information on the number of non-principal prime ideals in an ar...

This paper explores some foundational results of algebraic number theory. We focus on Dedekind rings and unique factorization of prime ideals, as well as some celebrated consequences such as a partial proof of Fermat’s last theorem due to Kummer, and the law of quadratic reciprocity.

An i?-module Mis a generator (of the category of modules) provided every module is a homomorphic image of a suitable direct sum of copies of M. Equivalently, some M has R as a summand. Except in the last section, all rings are assumed to be commutative, Noetherian domains, and modules are usually finitely generated. In this context generators are exactly those modules that have non-zero free su...

In 1977 Hartwig and Luh asked whether an element a in a Dedekind-finite ring R satisfying aR = a2R also satisfies Ra = Ra2. In this paper, we answer this question in the negative. We also prove that if a is an element of a Dedekind-finite exchange ring R and aR = a2R, then Ra = Ra2. This gives an easier proof of Dischinger’s theorem that left strongly π-regular rings are right strongly π-regula...

Given a direct sum G of cyclic groups, we find a sharp bound for the minimal number of proper subgroups whose union is G. This problem generalizes to sums of cyclic modules over more general rings, such as local and Artinian rings or Dedekind domains and reduces to covering vector spaces by proper subspaces. As a consequence, we are also able to solve the analogous problem for monoids.

The aim of this paper is to investigate generalizations locally artinian supplemented modules in module theory, namely radical and strongly modules. We have obtained elementary features for them. Also, we characterized by left perfect rings. Finally, proved that the reduced part a $R$-module has same property over Dedekind domain $R$.