scholarly journals The Relation between Almost Noetherian Module, Almost Finitely Generated Module and T-Noetherian Module

2019 ◽  
Vol 1306 ◽  
pp. 012001
Author(s):  
A Faisol ◽  
B Surodjo ◽  
S Wahyuni
Keyword(s):  
Finitely Generated ◽  
Finitely Generated Module ◽  
Noetherian Module

scholarly journals The asymptotic closure of an ideal relative to a module

2001 ◽  
Vol 32 (3) ◽  
pp. 231-235
Author(s):  
Sylvia M. Foster ◽  
Johnny A. Johnson
Keyword(s):  
Commutative Ring ◽  
Integral Closure ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Noetherian Module ◽  
The Ideal

In this paper we introduce the concept of the asymptotic closure of an ideal of a commutative ring $ R $ with identity relative to a unitary $ R $-module $ M $. This work extends results from P. Samuel, M. Nagata, J. W. Petro and Sharp, Tiras, and Yassi. Our objectives in this paper are to establish the cancellation law for the asymptotic completion of an ideal relative to a finitely generated module and show that the integral closure of an ideal relative to a Noetherian module $ M $ coincides with the asymptotic closure of the ideal relative to the Noetherian module $ M $.

scholarly journals Decompositions of modules into projective modules and CS-modules

2000 ◽  
Vol 62 (1) ◽  
pp. 159-164
Author(s):  
Somyot Plubtieng
Keyword(s):  
Projective Module ◽  
Injective Module ◽  
Projective Modules ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Noetherian Module ◽  
Direct Sum ◽  
Continuous Module

Let M be a right R-module. It is shown that M is a locally Noetherian module if every finitely generated module in σ[M] is a direct sum of a projective module and a CS-module. Moreover, if every module in σ[M] is a direct sum of a projective module and a CS-module, then every module in σ[M] is a direct sum of modules which are either indecomposable projective or uniform Σ-quasi-injective. In particular, if every module in σ[M] is a direct sum of a projective module and a quasi-continuous module, then every module in σ[M] is a direct sum of a projective module and a quasi-injective module.

On a Class of Automaton Algebras

2016 ◽  
Vol 60 (1) ◽  
pp. 31-38 ◽  
Author(s):  
Ferran Cedó ◽  
Jan Okniński
Keyword(s):  
Finitely Generated ◽  
Finitely Generated Module ◽  
Commutative Subalgebra

AbstractWe show that every finitely generated algebra that is a finitely generated module over a finitely generated commutative subalgebra is an automaton algebra in the sense of Ufnarovskii.

scholarly journals Gelfand-Kirillov Dimensions of Modules over Differential Difference Algebras

2016 ◽  
Vol 23 (04) ◽  
pp. 701-720 ◽  
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.

Prime uniserial modules and rings

2019 ◽  
Vol 18 (02) ◽  
pp. 1950035 ◽  
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?”.

Complete Intersection Flat Dimension and the Intersection Theorem

2012 ◽  
Vol 19 (spec01) ◽  
pp. 1161-1166
Author(s):  
Parviz Sahandi ◽  
Tirdad Sharif ◽  
Siamak Yassemi
Keyword(s):  
Local Ring ◽  
Complete Intersection ◽  
Intersection Theorem ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Gorenstein Dimension ◽  
Flat Dimension ◽  
Gorenstein Flat ◽  
Similar Extension

Any finitely generated module M over a local ring R is endowed with a complete intersection dimension CI-dim RM and a Gorenstein dimension G-dim RM. The Gorenstein dimension can be extended to all modules over the ring R. This paper presents a similar extension for the complete intersection dimension, and mentions the relation between this dimension and the Gorenstein flat dimension. In addition, we show that in the intersection theorem, the flat dimension can be replaced by the complete intersection flat dimension.

scholarly journals Torsion in tensor powers of modules

2015 ◽  
Vol 219 ◽  
pp. 113-125
Author(s):  
Olgur Celikbas ◽  
Srikanth B. Iyengar ◽  
Greg Piepmeyer ◽  
Roger Wiegand
Keyword(s):  
Local Ring ◽  
Complete Intersection ◽  
Tensor Products ◽  
Central Theme ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Free Modules ◽  
Tensor Powers ◽  
Torsion Free ◽  
Nonzero Torsion

AbstractTensor products usually have nonzero torsion. This is a central theme of Auslander's 1961 paper; the theme continues in the work of Huneke and Wiegand in the 1990s. The main focus in this article is on tensor powers of a finitely generated module over a local ring. Also, we study torsion-free modulesNwith the property thatM ⊗RNhas nonzero torsion unlessMis very special. An important example of such a moduleNis the Frobenius powerpeRover a complete intersection domainRof characteristicp> 0.

scholarly journals Finitely generated cyclic extensions of free groups are residually finite

1971 ◽  
Vol 5 (1) ◽  
pp. 87-94 ◽  
Author(s):  
Gilbert Baumslag
Keyword(s):  
Free Group ◽  
Principal Ideal ◽  
Free Groups ◽  
Cyclic Extension ◽  
Principal Ideal Domain ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Residually Finite ◽  
Ideal Domain ◽  
Direct Sum

We establish the result that a finitely generated cyclic extension of a free group is residually finite. This is done, in part, by making use of the fact that a finitely generated module over a principal ideal domain is a direct sum of cyclic modules.

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