we show  that a projective maximal submodule of afinitely generated, regular, extending module is a directsummand. hence, every finitely generated, regular, extendingmodule with projective maximal submodules is semisimple. as aconsequence, we observe that every regular, hereditary, extendingmodule is semisimple. this generalizes and simplifies a result of  dung and   smith. as another consequen...

An almost-direct product of free groups is an iterated semidirect product of finitely generated free groups in which the action of the constituent free groups on the homology of one another is trivial. We determine the structure of the cohomology ring of such a group. This is used to analyze the topological complexity of the associated Eilenberg-Mac Lane space. 1. Almost direct products of free...

Let D be a division ring with centre K and dim, D< ? a valuation on K and v a noninvariant extension of ? to D. We define the initial ramfication index of v over ?, ?(v/ ?) .Let A be a valuation ring of o with maximal ideal m, and v , v ,…, v noninvariant extensions of w to D with valuation rings A , A ,…, A . If B= A , it is shown that the following conditions are equivalent: (i) B i...

Abstract. In this article we revisit a problem regarding Bézout domains, namely, whether every Bézout domain is an elementary divisor domain. Elementary divisor domains where defined by Kaplansky [13] and generalized to rings with zero-divisors by Gillman and Henriksen [7]. Later, in [14] it was shown that a domain R is an elementary divisor domain if and only if every finitely presented R-modu...

It is shown that every commutative local ring of bounded module type is an almost maximal valuation ring. We say that an associative commutative unitary ring R is of bounded module type if there exists a positive integer n such that every finitely generated R-module is a direct sum of submodules generated by at most n elements. For instance, every Dedekind domain has bounded module type with bo...

A longstanding open problem in the theory of von Neumann regular rings is the question of whether every directly finite simple regular ring must be unit-regular. Recent work on this problem has been done by P. Menal, K.C . O'Meara, and the authors. To clarify some aspects of these new developments, we introduce and study the notion of almost isomorphism between finitely generated projective mod...

Elementary divisor domains were defined by Kaplansky [I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc. 66 (1949) 464–491] and generalized to rings with zero-divisors by Gillman and Henriksen [L. Gillman, M. Henriksen, Some remarks about elementary divisor rings, Trans. Amer. Math. Soc. 82 (1956) 362–365]. In [M.D. Larsen, W.J. Lewis, T.S. Shores, Elementary divisor rings a...