scholarly journals Decompositions of modules over a discrete valuation ring

1979 ◽  
Vol 27 (3) ◽  
pp. 284-288 ◽  
Author(s):  
Robert O. Stanton
Keyword(s):  
Direct Summand ◽  
Valuation Ring ◽  
Discrete Valuation Ring ◽  
Torsion Module ◽  
Direct Sum ◽  
Rank One ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free

AbstractLet N be a direct summand of a module which is a direct sum of modules of torsion-free rank one over a discrete valuation ring. Then there is a torsion module T such that N⊕T is also a direct sum of modules of torsion-free rank one.

scholarly journals ON PROJECTIVELY INERT SUBGROUPS OF COMPLETELY DECOMPOSABLE FINITE RANK GROUPS

2020 ◽  
pp. 63-68
Author(s):  
Andrey R. CHEKHLOV ◽  
◽  
Olesya V. IVANETS ◽  
Keyword(s):  
Prime Number ◽  
Finite Rank ◽  
Invariant Subgroup ◽  
Direct Sum ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free

Let a group G be a finite direct sum of torsion-free rank 1 groups Gi. It is proved that every projectively inert subgroup of G is commensurate with a fully invariant subgroup if and only if all Gi are not divisible by any prime number p, and for different subgroups Gi and Gj their types are either equal or incomparable.

D3-MODULES VERSUS D4-MODULES – APPLICATIONS TO QUIVERS

2020 ◽  
pp. 1-27
Author(s):  
GABRIELLA D′ESTE ◽  
DERYA KESKİN TÜTÜNCÜ ◽  
RACHID TRIBAK
Keyword(s):  
Direct Summand ◽  
Valuation Ring ◽  
Discrete Valuation Ring ◽  
Dedekind Domain ◽  
Projective Modules ◽  
Direct Sum ◽  
Cyclic Submodules ◽  
Rickart Modules

Abstract A module M is called a D4-module if, whenever A and B are submodules of M with M = A ⊕ B and f : A → B is a homomorphism with Imf a direct summand of B, then Kerf is a direct summand of A. The class of D4-modules contains the class of D3-modules, and hence the class of semi-projective modules, and so the class of Rickart modules. In this paper we prove that, over a commutative Dedekind domain R, for an R-module M which is a direct sum of cyclic submodules, M is direct projective (equivalently, it is semi-projective) iff M is D3 iff M is D4. Also we prove that, over a prime PI-ring, for a divisible R-module X, X is direct projective (equivalently, it is Rickart) iff X ⊕ X is D4. We determine some D3-modules and D4-modules over a discrete valuation ring, as well. We give some relevant examples. We also provide several examples on D3-modules and D4-modules via quivers.

scholarly journals I-Rad-⊕-supplemented modules

2019 ◽  
Vol 11 (1) ◽  
pp. 224-233
Author(s):  
Burcu Nişancı Türkmen
Keyword(s):  
Direct Summand ◽  
Valuation Ring ◽  
Discrete Valuation Ring ◽  
Direct Sum

Abstract Let M be an R-module and I be an ideal of R. We say that M is I-Rad-⊕-supplemented, provided for every submodule N of M, there exists a direct summand K of M such that M = N + K, N ∩ K ⊆ IK and N ∩ K Rad(K). The aim of this paper is to show new properties of I-Rad-⊕-supplemented modules. Especially, we show that any finite direct sum of I-Rad-⊕-supplemented modules is I-Rad-⊕-supplemented. We also prove that an R-module M is I-Rad-⊕-supplemented if and only if K and ${M \over K}$ are I-Rad-⊕-supplemented for a fully invariant direct summand K of M. Finally, we determine the structure of I-Rad-⊕-supplemented modules over a discrete valuation ring.

scholarly journals Torsion-free rank one sheaves over del Pezzo orders

2018 ◽  
Vol 493 ◽  
pp. 251-266
Author(s):  
Norbert Hoffmann ◽  
Fabian Reede
Keyword(s):  
Rank One ◽  
Free Rank ◽  
Del Pezzo ◽  
Torsion Free Rank ◽  
Torsion Free

MODULAR ABELIAN GROUP ALGEBRAS

2010 ◽  
Vol 03 (02) ◽  
pp. 275-293
Author(s):  
Peter Danchev
Keyword(s):  
Abelian Group ◽  
Factor Group ◽  
Compact Groups ◽  
Arbitrary Group ◽  
P Elements ◽  
Rank One ◽  
Free Rank ◽  
Torsion Groups ◽  
Torsion Free Rank ◽  
Torsion Free

Suppose FG is the F-group algebra of an arbitrary multiplicative abelian group G with p-component of torsion Gp over a field F of char (F) = p ≠ 0. Our theorems state thus: The factor-group S(FG)/Gp of all normed p-units in FG modulo Gp is always totally projective, provided G is a coproduct of groups whose p-components are of countable length and F is perfect. Moreover, if G is a p-mixed coproduct of groups with torsion parts of countable length and FH ≅ FG as F-algebras, then there is a totally projective p-group T of length ≤ Ω such that H × T ≅ G × T. These are generalizations to results by Hill-Ullery (1997). As a consequence, if G is a p-splitting coproduct of groups each of which has p-component with length < Ω and FH ≅ FG are F-isomorphic, then H is p-splitting. This is an extension of a result of May (1989). Our applications are the following: Let G be p-mixed algebraically compact or p-mixed splitting with torsion-complete Gp or p-mixed of torsion-free rank one with torsion-complete Gp. Then the F-isomorphism FH ≅ FG for any group H implies H ≅ G. Moreover, letting G be a coproduct of torsion-complete p-groups or G be a coproduct of p-local algebraically compact groups, then [Formula: see text]-isomorphism [Formula: see text] for an arbitrary group H over the simple field [Formula: see text] of p-elements yields H ≅ G. These completely settle in a more general form a question raised by May (1979) for p-torsion groups and also strengthen results due to Beers-Richman-Walker (1983).

scholarly journals Finite torsion-free rank endomorphism rings

2015 ◽  
Vol 31 (1) ◽  
pp. 39-43
Author(s):  
SIMION BREAZ ◽  
Keyword(s):  
Abelian Group ◽  
Small Group ◽  
Endomorphism Ring ◽  
Endomorphism Rings ◽  
Direct Sum ◽  
Bounded Group ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free

We prove that a finite torsion-free rank abelian group with finite torsion-free rank endomorphism ring is a direct sum of a bounded group and a self-small group.

scholarly journals On mixed groups of torsion-free rank one

1967 ◽  
Vol 11 (1) ◽  
pp. 134-144 ◽  
Author(s):  
Charles K. Megibben
Keyword(s):  
Rank One ◽  
Free Rank ◽  
Mixed Groups ◽  
Torsion Free Rank ◽  
Torsion Free
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