Finitely Generated Modules over Principal Ideal Domains

1962 ◽  
Vol 69 (5) ◽  
pp. 398
Author(s):  
W. J. Wong
Keyword(s):  
Principal Ideal ◽  
Finitely Generated ◽  
Finitely Generated Modules ◽  
Principal Ideal Domains

scholarly journals Torsion and protorsion modules over free ideal rings

1970 ◽  
Vol 11 (4) ◽  
pp. 490-498
Author(s):  
P. M. Cohn
Keyword(s):  
Principal Ideal ◽  
Finitely Generated ◽  
Commutative Case ◽  
Principal Ideal Domains ◽  
Torsion Modules ◽  
Noncommutative Analogue

Free ideal rings (or firs, cf. [2, 3] and § 2 below) form a noncommutative analogue of principal ideal domains, to which they reduce in the commutative case, and in [3] a category TR of right R-modules was defined, over any fir R, which forms an analogue of finitely generated torsion modules. The category TR was shown to be abelian, and all its objects have finite composition length; more over, the corresponding category RT of left R-modules is dual to TR.

Strongly D2 modules and their applications

2021 ◽  
pp. 2250071
Author(s):  
Mingzhao Chen ◽  
Hwankoo Kim ◽  
Fanggui Wang
Keyword(s):  
Positive Integer ◽  
Principal Ideal ◽  
Projective Module ◽  
Negative Answer ◽  
Principal Ideal Domain ◽  
Finitely Generated ◽  
Ideal Domain ◽  
Finitely Generated Modules ◽  
Structural Characterizations ◽  
Semihereditary Rings

An [Formula: see text]-module [Formula: see text] is called strongly [Formula: see text] if [Formula: see text] is a [Formula: see text] (equivalently, direct projective) module for every positive integer [Formula: see text]. In this paper, we consider the class of quasi-projective [Formula: see text]-modules, the class of strongly [Formula: see text] [Formula: see text]-modules and the class of [Formula: see text]-modules. We first show that these classes are distinct, which gives a negative answer to the question raised by Li–Chen–Kourki. We also give structural characterizations of strongly [Formula: see text] modules for finitely generated modules over a principal ideal domain. In addition, we characterize some rings such as Artinian semisimple rings, hereditary rings, semihereditary rings and perfect rings in terms of strongly [Formula: see text] modules.

On the Criteria of D.D. Anderson for Invertible and Flat Ideals

1986 ◽  
Vol 29 (1) ◽  
pp. 25-32 ◽  
Author(s):  
David E. Dobbs
Keyword(s):  
Principal Ideal ◽  
Integral Domain ◽  
Nonzero Ideal ◽  
Finitely Generated ◽  
Integral Domains ◽  
Maximal Ideals ◽  
Dedekind Domains ◽  
Principal Ideal Domains ◽  
Bezout Domains ◽  
Torsion Free

AbstractLet R be an integral domain. It is proved that if a nonzero ideal I of R can be generated by n < ∞ elements, then I is invertible (i.e., flat) if and only if I(∩ Rai) = ∩ Iai for all { a1, . . ., a n﹜ ⊂ I. The article's main focus is on torsion-free R-modules E which are LCM-stable in the sense that E(Ra ∩ Rb) = Ea ∩ Eb for all a, b ∈ R. By means of linear relations, LCM-stableness is shown to be equivalent to a weak aspect of flatness. Consequently, if each finitely generated ideal of R may be 2-generated, then each LCM-stable R-module is flat. Finally, LCM-stableness of maximal ideals serves to characterize Prüfer domains, Dedekind domains, principal ideal domains, and Bézout domains amongst suitably larger classes of integral domains.

scholarly journals Rings which are nearly principal ideal domains

1998 ◽  
Vol 40 (3) ◽  
pp. 343-351 ◽  
Author(s):  
A. W. Chatters
Keyword(s):  
Prime Number ◽  
Principal Ideal ◽  
Projective Modules ◽  
Finitely Generated ◽  
Principal Ideal Domains

We study a class of rings which are closely related to principal ideal domains, and prove in particular that finitely-generated projective modules over such rings are free. Examples include the ring of Lipschitz quaternions; Z[a½] with d = —3 or d = —7; and any subring R of M2(Z) such that R ⊇ M2(pZ) for some prime number/? and R/M2(pZ) is a field with p2 elements.

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.

Finitely generated modules over valuation rings

1984 ◽  
Vol 12 (15) ◽  
pp. 1795-1812 ◽  
Author(s):  
Luigi Salce ◽  
Paolo Zanardo
Keyword(s):  
Finitely Generated ◽  
Valuation Rings ◽  
Finitely Generated Modules

scholarly journals Maximal depth property of finitely generated modules

2018 ◽  
Vol 17 (11) ◽  
pp. 1850202 ◽  
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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