Stable equivalence and rings whose modules are a direct sum of finitely generated modules

AbstractWe study direct-sum decompositions of torsion-free, finitely generated modules over a (commutative) Bass ring R through the factorization theory of the corresponding monoid T(R). Results of Levy–Wiegand and Levy–Odenthal together with a study of the local case yield an explicit description of T(R). The monoid is typically neither factorial nor cancellative. Nevertheless, we construct a transfer homomorphism to a monoid of graph agglomerations—a natural class of monoids serving as combinatorial models for the factorization theory of T(R). As a consequence, the monoid T(R) is transfer Krull of finite type and several finiteness results on arithmetical invariants apply. We also establish results on the elasticity of T(R) and characterize when T(R) is half-factorial. (Factoriality, that is, torsion-free Krull–Remak–Schmidt–Azumaya, is characterized by a theorem of Levy–Odenthal.) The monoids of graph agglomerations introduced here are also of independent interest.

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Weakly Projective and Weakly Injective Modules

Canadian Journal of Mathematics ◽

10.4153/cjm-1994-055-9 ◽

1994 ◽

Vol 46 (5) ◽

pp. 971-981 ◽

Cited By ~ 3

Author(s):

S. K. Jain ◽

S. R. López-Permouth ◽

K. Oshiro ◽

M. A. Saleh

Keyword(s):

Positive Integer ◽

Artinian Ring ◽

Projective Cover ◽

Finitely Generated ◽

Matrix Rings ◽

Direct Sum ◽

Injective Modules ◽

Qf Ring ◽

Finitely Generated Modules

AbstractA module M is said to be weakly N-projective if it has a projective cover π: P(M) ↠M and for each homomorphism : P(M) → N there exists an epimorphism σ:P(M) ↠M such that (kerσ) = 0, equivalently there exists a homomorphism :M ↠N such that σ= . A module M is said to be weakly projective if it is weakly N-projective for all finitely generated modules N. Weakly N-injective and weakly injective modules are defined dually. In this paper we study rings over which every weakly injective right R-module is weakly projective. We also study those rings over which every weakly projective right module is weakly injective. Among other results, we show that for a ring R the following conditions are equivalent:(1) R is a left perfect and every weakly projective right R-module is weakly injective.(2) R is a direct sum of matrix rings over local QF-rings.(3) R is a QF-ring such that for any indecomposable projective right module eR and for any right ideal I, soc(eR/eI) = (eR/eJ)n for some positive integer n.(4) R is right artinian ring and every weakly injective right R-module is weakly projective.(5) Every weakly projective right R-module is weakly injective and every weakly injective right R-module is weakly projective.

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Gelfand-Kirillov Dimensions of Modules over Differential Difference Algebras

Algebra Colloquium ◽

10.1142/s1005386716000596 ◽

2016 ◽

Vol 23 (04) ◽

pp. 701-720 ◽

Cited By ~ 2

Author(s):

Xiangui Zhao ◽

Yang Zhang

Keyword(s):

Computational Method ◽

Finitely Generated ◽

Finitely Generated Module ◽

Polynomial Algebras ◽

Ore Extensions ◽

Basis Method ◽

Finitely Generated Modules ◽

Kirillov Dimension

Differential difference algebras are generalizations of polynomial algebras, quantum planes, and Ore extensions of automorphism type and of derivation type. In this paper, we investigate the Gelfand-Kirillov dimension of a finitely generated module over a differential difference algebra through a computational method: Gröbner-Shirshov basis method. We develop the Gröbner-Shirshov basis theory of differential difference algebras, and of finitely generated modules over differential difference algebras, respectively. Then, via Gröbner-Shirshov bases, we give algorithms for computing the Gelfand-Kirillov dimensions of cyclic modules and finitely generated modules over differential difference algebras.

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Prime uniserial modules and rings

Journal of Algebra and Its Applications ◽

10.1142/s021949881950035x ◽

2019 ◽

Vol 18 (02) ◽

pp. 1950035 ◽

Cited By ~ 1

Author(s):

M. Behboodi ◽

Z. Fazelpour

Keyword(s):

Commutative Rings ◽

Finitely Generated ◽

Finitely Generated Module ◽

Direct Sum ◽

Dedekind Domains ◽

Prime Submodules ◽

Uniserial Modules ◽

Structure Formula

We define prime uniserial modules as a generalization of uniserial modules. We say that an [Formula: see text]-module [Formula: see text] is prime uniserial ([Formula: see text]-uniserial) if its prime submodules are linearly ordered by inclusion, and we say that [Formula: see text] is prime serial ([Formula: see text]-serial) if it is a direct sum of [Formula: see text]-uniserial modules. The goal of this paper is to study [Formula: see text]-serial modules over commutative rings. First, we study the structure [Formula: see text]-serial modules over almost perfect domains and then we determine the structure of [Formula: see text]-serial modules over Dedekind domains. Moreover, we discuss the following natural questions: “Which rings have the property that every module is [Formula: see text]-serial?” and “Which rings have the property that every finitely generated module is [Formula: see text]-serial?”.

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Finitely generated modules over valuation rings

Communications in Algebra ◽

10.1080/00927878408823083 ◽

1984 ◽

Vol 12 (15) ◽

pp. 1795-1812 ◽

Cited By ~ 12

Author(s):

Luigi Salce ◽

Paolo Zanardo

Keyword(s):

Finitely Generated ◽

Valuation Rings ◽

Finitely Generated Modules

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Maximal depth property of finitely generated modules

Journal of Algebra and Its Applications ◽

10.1142/s021949881850202x ◽

2018 ◽

Vol 17 (11) ◽

pp. 1850202 ◽

Cited By ~ 1

Author(s):

Ahad Rahimi

Keyword(s):

Local Ring ◽

Finitely Generated ◽

Noetherian Local Ring ◽

Maximal Depth ◽

Finitely Generated Modules ◽

Associated Prime

Let [Formula: see text] be a Noetherian local ring and [Formula: see text] a finitely generated [Formula: see text]-module. We say [Formula: see text] has maximal depth if there is an associated prime [Formula: see text] of [Formula: see text] such that depth [Formula: see text]. In this paper, we study finitely generated modules with maximal depth. It is shown that the maximal depth property is preserved under some important module operations. Generalized Cohen–Macaulay modules with maximal depth are classified. Finally, the attached primes of [Formula: see text] are considered for [Formula: see text].

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A note on the fixed subring of an FPF ring

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700003543 ◽

1989 ◽

Vol 40 (1) ◽

pp. 109-111 ◽

Cited By ~ 3

Author(s):

John Clark

Keyword(s):

Finite Group ◽

Associative Ring ◽

Finitely Generated ◽

Group Of Automorphisms ◽

Direct Sum ◽

Epimorphic Image ◽

A Finite Group

An associative ring R with identity is called a left (right) FPF ring if given any finitely generated faithful left (right) R-module A and any left (right) R-module M then M is the epimorphic image of a direct sum of copies of A. Faith and Page have asked if the subring of elements fixed by a finite group of automorphisms of an FPF ring need also be FPF. Here we present examples showing the answer to be negative in general.

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