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 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.

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 ABELIAN GROUPS WITH A p2-BOUNDED SUBGROUP, REVISITED

2011 ◽  
Vol 10 (03) ◽  
pp. 377-389
Author(s):  
CARLA PETRORO ◽  
MARKUS SCHMIDMEIER
Keyword(s):  
Abelian Group ◽  
Prime Number ◽  
Maximal Ideal ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Uniserial Ring

Let Λ be a commutative local uniserial ring of length n, p be a generator of the maximal ideal, and k be the radical factor field. The pairs (B, A) where B is a finitely generated Λ-module and A ⊆B a submodule of B such that pmA = 0 form the objects in the category [Formula: see text]. We show that in case m = 2 the categories [Formula: see text] are in fact quite similar to each other: If also Δ is a commutative local uniserial ring of length n and with radical factor field k, then the categories [Formula: see text] and [Formula: see text] are equivalent for certain nilpotent categorical ideals [Formula: see text] and [Formula: see text]. As an application, we recover the known classification of all pairs (B, A) where B is a finitely generated abelian group and A ⊆ B a subgroup of B which is p2-bounded for a given prime number p.

scholarly journals On p-Separability of Subgroups of Free Metabelian Groups

2006 ◽  
Vol 13 (02) ◽  
pp. 289-294 ◽  
Author(s):  
Valerij G. Bardakov
Keyword(s):  
Prime Number ◽  
Cyclic Group ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Metabelian Groups ◽  
Isolated Subgroup

We prove that every free metabelian non-cyclic group has a finitely generated isolated subgroup which is not separable in the class of nilpotent groups. As a corollary, we prove that for every prime number p, an arbitrary free metabelian non-cyclic group has a finitely generated p′-isolated subgroup which is not p-separable.

scholarly journals On finitely generated n-SG-projective modules

2010 ◽  
Vol 121 (1) ◽  
pp. 35-44
Author(s):  
Driss Bennis
Keyword(s):  
Projective Modules ◽  
Finitely Generated
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