T-idempotent invariant modules

2019 ◽  
Vol 18 (06) ◽  
pp. 1950115 ◽  
Author(s):  
Shahabaddin Ebrahimi Atani ◽  
Mehdi Khoramdel ◽  
Saboura Dolati Pish Hesari
Keyword(s):  
Injective Envelope ◽  
Finitely Generated ◽  
Idempotent Endomorphism

We introduce and investigate [Formula: see text]-idempotent invariant modules. We call an endomorphism [Formula: see text] of [Formula: see text], a [Formula: see text]-idempotent endomorphism if [Formula: see text] defined by [Formula: see text] is an idempotent and we call a module [Formula: see text] is [Formula: see text]-idempotent invariant, if it is invariant under [Formula: see text]-idempotents of its injective envelope. We prove a module [Formula: see text] is [Formula: see text]-idempotent invariant if and only if [Formula: see text], [Formula: see text] is quasi-injective, [Formula: see text] is quasi-continuous and [Formula: see text] is [Formula: see text]-injective. The class of rings [Formula: see text] for which every (finitely generated, cyclic, free) [Formula: see text]-module is [Formula: see text]-idempotent invariant is characterized. Moreover, it is proved that if [Formula: see text] is right q.f.d., then every [Formula: see text]-idempotent invariant [Formula: see text]-module is quasi-injective exactly when every nonsingular uniform [Formula: see text]-module is quasi-injective.

Neat-phantom and clean-cophantom morphisms

2020 ◽  
pp. 2150172
Author(s):  
Lixin Mao
Keyword(s):  
Exact Sequence ◽  
Projective Resolution ◽  
Injective Envelope ◽  
Finitely Generated ◽  
The Relationship ◽  
Exact Sequences

Let [Formula: see text] be the class of all left [Formula: see text]-modules [Formula: see text] which has a projective resolution by finitely generated projectives. An exact sequence [Formula: see text] of right [Formula: see text]-modules is called neat if the sequence [Formula: see text] is exact for any [Formula: see text]. An exact sequence [Formula: see text] of left [Formula: see text]-modules is called clean if the sequence [Formula: see text] is exact for any [Formula: see text]. We prove that every [Formula: see text]-module has a clean-projective precover and a neat-injective envelope. A morphism [Formula: see text] of right [Formula: see text]-modules is called a neat-phantom morphism if [Formula: see text] for any [Formula: see text]. A morphism [Formula: see text] of left [Formula: see text]-modules is said to be a clean-cophantom morphism if [Formula: see text] for any [Formula: see text]. We establish the relationship between neat-phantom (respectively, clean-cophantom) morphisms and neat (respectively, clean) exact sequences. Also, we prove that every [Formula: see text]-module has a neat-phantom cover with kernel neat-injective and a clean-cophantom preenvelope with cokernel clean-projective.

Modules which are invariant under nilpotents of their envelopes and covers

2020 ◽  
pp. 2150218
Author(s):  
Truong Cong Quynh ◽  
Adel Abyzov ◽  
Dinh Duc Tai
Keyword(s):  
Injective Envelope ◽  
Finitely Generated ◽  
Injective Modules ◽  
Semiprimary Ring ◽  
Nilpotent Endomorphism

A module is called nilpotent-invariant if it is invariant under any nilpotent endomorphism of its injective envelope [M. T. Koşan and T. C. Quynh, Nilpotent-invaraint modules and rings, Comm. Algebra 45 (2017) 2775–2782]. In this paper, we continue the study of nilpotent-invariant modules and analyze their relationship to (quasi-)injective modules. It is proved that a right module [Formula: see text] over a semiprimary ring is nilpotent-invariant iff all nilpotent endomorphisms of submodules of [Formula: see text] extend to nilpotent endomorphisms of [Formula: see text]. It is also shown that a right module [Formula: see text] over a prime right Goldie ring with [Formula: see text] is nilpotent-invariant iff it is injective. We also study nilpotent-coinvariant modules that are the dual notation of nilpotent-invariant modules. It is proved that if [Formula: see text] is a finitely generated nilpotent-coinvariant right module with [Formula: see text] square-full, then [Formula: see text] is quasi-projective. Some characterizations and structures of nilpotent-coinvariant modules are considered.

On semirings all of whose semimodules have injective envelopes

2020 ◽  
pp. 2150130
Author(s):  
S. N. Il’in
Keyword(s):  
Tensor Products ◽  
Injective Envelope ◽  
Finitely Generated ◽  
Converse Statement

It was shown in [Y. Katsov, Tensor products and injective envelopes of semimodules over additively regular semirings, Algebra Colloq. 4 (1997) 121–131.] that each semimodule over an additively regular semiring has an injective envelope. We show the converse statement is true as well; moreover, the result holds even if each finitely generated semimodule has an injective envelope.

Automorphisms of finite order of nilpotent groups II

2014 ◽  
Vol 51 (4) ◽  
pp. 547-555 ◽  
Author(s):  
B. Wehrfritz
Keyword(s):  
Nilpotent Group ◽  
Finite Order ◽  
Finite Index ◽  
Nilpotent Groups ◽  
Finitely Generated

Let G be a nilpotent group with finite abelian ranks (e.g. let G be a finitely generated nilpotent group) and suppose φ is an automorphism of G of finite order m. If γ and ψ denote the associated maps of G given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\gamma :g \mapsto g^{ - 1} \cdot g\phi and \psi :g \mapsto g \cdot g\phi \cdot g\phi ^2 \cdots \cdot \cdot g\phi ^{m - 1} for g \in G,$$ \end{document} then Gγ · kerγ and Gψ · ker ψ are both very large in that they contain subgroups of finite index in G.

Collectives of Automata in Finitely Generated Groups

2020 ◽  
Vol 108 (5-6) ◽  
pp. 671-678
Author(s):  
D. V. Gusev ◽  
I. A. Ivanov-Pogodaev ◽  
A. Ya. Kanel-Belov
Keyword(s):  
Finitely Generated ◽  
Finitely Generated Groups

scholarly journals TEST IDEALS IN RINGS WITH FINITELY GENERATED ANTI-CANONICAL ALGEBRAS – CORRIGENDUM

2016 ◽  
Vol 17 (4) ◽  
pp. 979-980
Author(s):  
Alberto Chiecchio ◽  
Florian Enescu ◽  
Lance Edward Miller ◽  
Karl Schwede
Keyword(s):  
Finitely Generated

Primitivity in representations of polycyclic groups

1980 ◽  
Vol 88 (1) ◽  
pp. 15-31 ◽  
Author(s):  
D. L. Harper
Keyword(s):  
Finite Group ◽  
Irreducible Representation ◽  
Nilpotent Group ◽  
Infinite Dimension ◽  
Finitely Generated ◽  
Polycyclic Groups

In an earlier paper (5) we showed that a finitely generated nilpotent group which is not abelian-by-finite has a primitive irreducible representation of infinite dimension over any non-absolute field. Here we are concerned primarily with the converse question: Suppose that G is a polycyclic-by-finite group with such a representation, then what can be said about G?

scholarly journals Embedding the Picard group inside the class group: the case of $$\mathbb {Q}$$-factorial complete toric varieties

2021 ◽  
Author(s):  
Michele Rossi ◽  
Lea Terracini
Keyword(s):  
Free Group ◽  
Toric Variety ◽  
Class Group ◽  
Abelian Groups ◽  
Toric Varieties ◽  
Picard Group ◽  
Closed Field ◽  
Finitely Generated ◽  
Free Part ◽  
Canonical Injection

AbstractLet X be a $$\mathbb {Q}$$ Q -factorial complete toric variety over an algebraic closed field of characteristic 0. There is a canonical injection of the Picard group $$\mathrm{Pic}(X)$$ Pic ( X ) in the group $$\mathrm{Cl}(X)$$ Cl ( X ) of classes of Weil divisors. These two groups are finitely generated abelian groups; while the first one is a free group, the second one may have torsion. We investigate algebraic and geometrical conditions under which the image of $$\mathrm{Pic}(X)$$ Pic ( X ) in $$\mathrm{Cl}(X)$$ Cl ( X ) is contained in a free part of the latter group.

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