Regularly weakly based modules over right perfect rings and Dedekind domains

A weak basis of a module is a generating set of the module minimal with respect to inclusion. A module is said to be regularly weakly based provided that each of its generating sets contains a weak basis. We study

1. (1)
   rings over which all modules are regularly weakly based, refining results of Nashier and Nichols, and
    
2. (2)
   regularly weakly based modules over Dedekind domains.
    

     

Injective and automorphism-invariant non-singular modules

Every automorphism-invariant non-singular right A-module is injective if and only if the factor ring of the ring A with respect to its right Goldie radical is a right strongly semiprime ring.     

OnE-compact modules over SISI rings

Summary We extend to the case of SISI rings, studied by Vamos in [8], the duality for noetherian rings studied by Orsatti in [7]. We prove that some properties of this duality can be generalized, while others hold only in the noetherian case.

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Riassunto Si estende al caso dei SISI anelli, studiati da Vamos in [8], la dualità studiata da Orsatti in [7] per gli anelli noetheriani. Si dimostra che alcune proprietà di questa dualità si possono generalizzare, mentre altre sono caratteristiche del caso noetheriano.

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Lavoro eseguito nell’ambito dell’attività dei gruppi di ricerca matematici del C.N.R.     

Quadratic modules over polynomial rings

One gives the complete solution of the problem of the reduction of quadratic spaces for the polynomial ring over a field.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 86, pp. 114–124, 1979.     

Flat modules over valuation rings

Let R be a valuation ring and let Q be its total quotient ring. It is proved that any singly projective (respectively flat) module is finitely projective if and only if Q is maximal (respectively artinian). It is shown that each singly projective module is a content module if and only if any non-unit of R is a zero-divisor and that each singly projective module is locally projective if and only if R is self-injective. Moreover, R is maximal if and only if each singly projective module is separable, if and only if any flat content module is locally projective. Necessary and sufficient conditions are given for a valuation ring with non-zero zero-divisors to be strongly coherent or π-coherent.A complete characterization of semihereditary commutative rings which are π-coherent is given. When R is a commutative ring with a self-FP-injective quotient ring Q, it is proved that each flat R-module is finitely projective if and only if Q is perfect.     

Torsion-free modules over primary rings

The Noetherian primary rings over which all indecomposable torsion-free modules in the sense of Bass have discrete normed endomorphism rings are described.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 57, pp. 100–116, 1976.     

Torsion-free modules over Dedekind rings

We obtain necessary and sufficient conditions in order that an arbitrary pure monoendomorphism of a module decomposed into a direct sum of rank 1 torsion-free modules over a Dedekind ring be an automorphism.     

Tilting modules over almost perfect domains

We provide a complete classification of all tilting modules and tilting classes over almost perfect domains, which generalizes the classifications of tilting modules and tilting classes over Dedekind and 1-Gorenstein domains. Assuming the APD is Noetherian, a complete classification of all cotilting modules is obtained (as duals of the tilting ones).     

Endomorphism rings of simple modules over group rings

If is a finitely generated nilpotent group which is not abelian-by-finite, a field, and a finite dimensional separable division algebra over , then there exists a simple module for the group ring with endomorphism ring . An example is given to show that this cannot be extended to polycyclic groups.

     

McCoy modules and related modules over commutative rings

Let M be a left R-module. Then M is a McCoy (resp., dual McCoy) module if for nonzero f(X)∈R[X] and m(X)∈M[X], f(X)m(X) = 0 implies there exists a nonzero r∈R (resp., m∈M) with rm(X) = 0 (resp., f(X)m = 0). We show that for R commutative every R-module is dual McCoy, but give an example of a non-McCoy module. A number of other results concerning (dual) McCoy modules as well as arithmetical, Gaussian, and Armendariz modules are given.     

Simple modules over generalized matrix rings

We study simple modules over the ring of generalized matrices. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 3, pp. 245–247, 2007.     

Pure projective modules and FP-injective modules over Morita rings

Let {\mathrm{\Lambda }}_{(0,0)}=\left(\begin{array}{cc}A& {}_A{N}_B\\ {}_B{N}_A& B\end{array}\right) be a Morita ring, where the bimodule homomorphisms \varphiand \psi are zero. We study the finite presentedness, locally coherence, pure projectivity, pure injectivity, and FP-injectivity of modules over {\mathrm{\Lambda }}_{(\mathrm{0},\mathrm{0})}. Some applications are then given.     

Differential projective modules over differential rings

Differential modules over a commutative differential ring which are projective as ring modules, with differential homomorphisms, form an additive category. Every projective ring module is shown occurs as the underlying module of a differential module. Differential modules, projective as ring modules, are shown to be direct summands of differential modules free as ring modules; those which are differential direct summands of differential direct sums of the ring being induced from the subring of constants. Every differential module finitely generated and projective as a ring module is shown to have this form after a faithfully flat finitely presented differential extension of the base.