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On Modules over Infinite Group Rings [article]

Gunnar Carlsson, Boris Goldfarb
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 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] .  ... 
arXiv:1509.02402v1 fatcat:ebor2r3th5hjhb5codqq5qp4vi

ON MODULES OVER GROUP RINGS OF GROUPS WITH RESTRICTIONS ON THE SYSTEM OF ALL PROPER SUBGROUPS

O.Yu. DASHKOVA
2013 Asian-European Journal of Mathematics  
We consider the class M of R-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.  ... 
doi:10.1142/s1793557113500319 fatcat:kqhmhxij6fa3xhra2ieoqgg4i4

On Schur Rings Over Infinite Groups [article]

Nicholas Bastian, Jaden Brewer, Andrew Misseldine
2018 arXiv   pre-print
This paper extends many classic results about Schur 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).  ... 
arXiv:1806.07010v1 fatcat:l2uqsn72l5di3aadvq37g23ube

Jónsson and HS Modules over Commutative Rings

Greg Oman
2014 International Journal of Mathematics and Mathematical Sciences  
LetRbe a commutative 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)?  ... 
doi:10.1155/2014/120907 fatcat:mgejz3yn4zfjpkiyizdy55wqqi

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.  ... 

On modules over group rings of groups with restrictions on the system of all proper subgroups [article]

O.Yu.Dashkova
2013 arXiv   pre-print
Let A be a 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.  ... 
arXiv:1305.0744v2 fatcat:coeiomzngneo5b2qsbtzgvkrge

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.  ... 

Modules over group rings of locally soluble groups with a certain condition of minimality

O. YU. Dashkova
2014 Miskolc Mathematical Notes  
Let A be an RG-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.  ... 
doi:10.18514/mmn.2014.1240 fatcat:zcgh4qo44fhdtlcolan4qbu37y

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  ... 

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  ... 

Group algebras whose simple modules are injective

Daniel R. Farkas, Robert L. Snider
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.  ... 
doi:10.1090/s0002-9947-1974-0357475-5 fatcat:fdkb2dpw4jcqlnr2hiipe54moe

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.  ... 

Metabelian groups with the same finite quotients

P.F. Pickel
1974 Bulletin of the Australian Mathematical Society  
We construct 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  ... 
doi:10.1017/s0004972700043689 fatcat:7qed7cnxinam3k62b2hsxjruiy

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.  ... 

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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