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On Modules over Infinite Group Rings
[article]

2015
*
arXiv
*
pre-print

Some results and constructions in this paper might be of independent interest as we learn to construct projective resolutions of finite type for certain

arXiv:1509.02402v1
fatcat:ebor2r3th5hjhb5codqq5qp4vi
*modules**over**group**rings*. ... When the*group**ring*is Noetherian there is a companion G-theory of R[Γ] which is often easier to compute. However, it is exceptionally rare that the*group**ring*is Noetherian for an*infinite**group*. ... There are systematic constructions of non-free stably free Z[Γ]*modules*for*infinite*discrete*groups*Γ, cf. section 17.4 of Johnson [25] . ...##
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ON MODULES OVER GROUP RINGS OF GROUPS WITH RESTRICTIONS ON THE SYSTEM OF ALL PROPER SUBGROUPS

2013
*
Asian-European Journal of Mathematics
*

We consider the class M of R-

doi:10.1142/s1793557113500319
fatcat:kqhmhxij6fa3xhra2ieoqgg4i4
*modules*where R is an associative*ring*. Let A be a*module**over*a*group**ring*RG, G be a*group*and let L(G) be the set of all proper subgroups of G. We suppose CG(A) = 1. ... We study the cases: 1) M is the class of all artinian R-*modules*, R is either the*ring*of integers or the*ring*of p-adic integers; 2) M is the class of all finite R-*modules*, R is an associative*ring*; 3) ...*Modules**over**group**rings*of*infinite**groups*have been considered recently in [13] . Artinian and noetherian*modules**over**group**rings*are a broad class of*modules**over**group**rings*. ...##
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On Schur Rings Over Infinite Groups
[article]

2018
*
arXiv
*
pre-print

This paper extends many classic results about Schur

arXiv:1806.07010v1
fatcat:l2uqsn72l5di3aadvq37g23ube
*rings*to the*infinite**groups*, including Leung-Man's classification of Schur*rings**over*finite cyclic*groups*. ... Schur*rings*are a type of subring of the*group**ring*that is spanned by a partition of the*group*that meets certain conditions. Past literature has exclusively focused*on*the finite*group*case. ... In general, the intersection of two Schur*modules*(*rings*)*over*an*infinite**group*need not be a Schur*module*(*ring*). ...##
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Jónsson and HS Modules over Commutative Rings

2014
*
International Journal of Mathematics and Mathematical Sciences
*

LetRbe a commutative

doi:10.1155/2014/120907
fatcat:mgejz3yn4zfjpkiyizdy55wqqi
*ring*with identity and letMbe an*infinite*unitaryR-*module*. (Unless indicated otherwise, all*rings*are commutative with identity 1≠0 and all*modules*are unitary.) ... In this survey paper, we bring the reader up to speed*on*current research*on*these structures by presenting the principal results*on*Jónsson and HS*modules*. ... The countably*infinite*Jónsson*modules*have been classified. Can*one*classify the uncountable Jónsson*modules**over*an arbitrary*ring*(or*over*some restricted class of*rings*, such as valuation*rings*)? ...##
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Page 604 of Mathematical Reviews Vol. 47, Issue 3
[page]

1974
*
Mathematical Reviews
*

Since integral

*group**rings*of finite*groups*are Gorenstein orders, this gives the injective*modules**over*integral*group**rings*. ... W. 3453 Injective*modules*for*group**rings*and Gorenstein orders. J. Algebra 24 (1973), 465-472. The author classifies injective*modules**over*Gorenstein orders. ...##
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On modules over group rings of groups with restrictions on the system of all proper subgroups
[article]

2013
*
arXiv
*
pre-print

Let A be a

arXiv:1305.0744v2
fatcat:coeiomzngneo5b2qsbtzgvkrge
*module**over*a*group**ring*RG where G is a*group*and let L(G) be a set of all proper subgroups of G such that if H ∈ L(G) then A/C_A(H) belongs to M. ... a locally soluble*group*; 3) M is a class of all finite R--*modules*, R is an associative*ring*. ...*Modules**over**group**rings*of*infinite**groups*have been considered recently in [6] . Artinian and noetherian*modules**over**group**rings*are a broad class of*modules**over**group**rings*. ...##
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Page 697 of Mathematical Reviews Vol. , Issue 92b
[page]

1992
*
Mathematical Reviews
*

The author shows how to construct

*modules**over*a local stationary algebra using a graded graph and illustrates this for the*infinite*symmetric*group*S.. with the*infinite*Young lattice. ... Results*on*coherence, semi-heredity and injectivity for*group**rings*and associated finitely generated*modules*are obtained or gener- alised to cover finitely presented*modules*. ...##
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Modules over group rings of locally soluble groups with a certain condition of minimality

2014
*
Miskolc Mathematical Notes
*

Let A be an RG-

doi:10.18514/mmn.2014.1240
fatcat:zcgh4qo44fhdtlcolan4qbu37y
*module*, where R is an associative*ring*, A=C A .G/ is an*infinite*Rmodule, C G .A/ D 1, G is a locally soluble*group*. ... Let L nf .G/ be the system of all subgroups H Ä G such that quotient*modules*A=C A .H / are*infinite*R-*modules*. ... Study of*modules**over**group**rings*of*infinite**groups*requires some additional restrictions as in the case of*infinite*dimensional linear*groups*. ...##
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Page 6146 of Mathematical Reviews Vol. , Issue 2000i
[page]

2000
*
Mathematical Reviews
*

A constructive description is given for the

*groups*of units of finitely generated*rings*of matrices*over*the field of rational numbers. ... The authors characterize quasi-directly*infinite**rings*as those*rings*containing as subrings certain homomorphic images of a*ring*@ of N x N matrices which is generated*over*the*ring*of integers by two ...##
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Page 3523 of Mathematical Reviews Vol. , Issue 97F
[page]

1997
*
Mathematical Reviews
*

(S-MELB; Parkville)

*Modules**over*nilpotent*group**rings*. J. London Math. Soc. (2) 52 (1995), no. 3, 467-481. ... Such*modules*are important in the representation the- ory of finitely generated nilpotent*groups*, for example, since every non-zero*module**over*a*group**ring*of such a*group*contains a non- zero submodule ...##
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Group algebras whose simple modules are injective

1974
*
Transactions of the American Mathematical Society
*

*modules*are finite dimensional

*over*their commuting

*rings*. ... The condition that all simple

*modules*

*over*a

*ring*be injective first appeared in a theorem due to Kaplansky: a commutative

*ring*satisfies the condition if and only if it is von Neumann regular. ...

*One*direction has already been proved in Lemma 3. Thus we assume A has a simple

*module*

*infinite*dimensional

*over*its commuting

*ring*and construct a simple

*module*which is not injective. ...

##
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Page 5867 of Mathematical Reviews Vol. , Issue 2003h
[page]

2003
*
Mathematical Reviews
*

A proof of the main result is based

*on*the full description of Cayley*rings*introduced in this article and normal Schur*rings**over*a finite cyclic*group*. ... (UKR-DNEP; Dnepropetrovsk Minimally*infinite**modules**over*metabelian*groups*of finite rank. (Russian. Russian summary) Mat. Sb. 193 (2002), no. 5, 129-148. Let R be a*ring*. ...##
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Metabelian groups with the same finite quotients

1974
*
Bulletin of the Australian Mathematical Society
*

We construct

doi:10.1017/s0004972700043689
fatcat:7qed7cnxinam3k62b2hsxjruiy
*infinitely*many nonisomorphic finitely presented metabelian*groups*with the same finite quotients, using*modules**over*a suitably chosen*ring*. ... The Picard*group*Pic (if) of if is the*group*of isomorphism classes of rank*one*projective*modules**over*R , under tensor product.available at https://www.cambridge.org/core/terms. https://doi.org/10.be ...##
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Page 6654 of Mathematical Reviews Vol. , Issue 89M
[page]

1989
*
Mathematical Reviews
*

Bergman, “

*Groups*acting*on**rings*,*group*graded*rings*, and beyond”, Preprint; per bibl.; “*On*Jacobson radicals of graded*rings*”, Preprint; per bibl.] may be extended to the case of*infinite**groups*by associating ... to every*group*-graded*ring*R, a*ring*R#G* (without identity if G is*infinite*) such that G acts as a*group*of automorphisms*on*R#G* and the duality theorems of M. ...##
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Page 140 of Mathematical Reviews Vol. , Issue 2002A
[page]

2002
*
Mathematical Reviews
*

Moreover, the

*group*B*on**infinite*strings acts*on*the set of exceptional sequences of*infinite*length but the action may not be transitive. ... serial*modules**over*an arbitrary associative*ring*R with identity lp 40g. ...
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