A Note on Compactifying Artinian Rings

1974 ◽  
Vol 26 (3) ◽  
pp. 580-582 ◽  
Author(s):  
David K. Haley
Keyword(s):  
Finite Group ◽  
Artinian Ring ◽  
Topological Ring ◽  
Artinian Rings ◽  
Direct Sum ◽  
A Finite Group

In this note a number of compactifications are discussed within the class of artinian rings. In [1] the following was proved:Theorem. For an artinian ring R the following are equivalent:(1) R is equationally compact.(2) R+ ≃ B ⊕ P, where B is a finite group, P is a finite direct sum of Prüfer groups, and R · P = P · R = {0}.(3) R is a retract of a compact topological ring.

scholarly journals A note on the fixed subring of an FPF ring

1989 ◽  
Vol 40 (1) ◽  
pp. 109-111 ◽  
Author(s):  
John Clark
Keyword(s):  
Finite Group ◽  
Associative Ring ◽  
Finitely Generated ◽  
Group Of Automorphisms ◽  
Direct Sum ◽  
Epimorphic Image ◽  
A Finite Group

An associative ring R with identity is called a left (right) FPF ring if given any finitely generated faithful left (right) R-module A and any left (right) R-module M then M is the epimorphic image of a direct sum of copies of A. Faith and Page have asked if the subring of elements fixed by a finite group of automorphisms of an FPF ring need also be FPF. Here we present examples showing the answer to be negative in general.

Rings whose Indecomposable Injective Modules are Uniserial

1982 ◽  
Vol 34 (4) ◽  
pp. 797-805 ◽  
Author(s):  
David A. Hill
Keyword(s):  
Artinian Ring ◽  
Dimensional Algebra ◽  
Artinian Rings ◽  
Direct Sum ◽  
Injective Modules ◽  
Finite Dimensional ◽  
Uniserial Modules ◽  
Left And Right ◽  
Finitely Generated Modules ◽  
Algebra Over A Field

A module is uniserial in case its submodules are linearly ordered by inclusion. A ring R is left (right) serial if it is a direct sum of uniserial left (right) R-modules. A ring R is serial if it is both left and right serial. It is well known that for artinian rings the property of being serial is equivalent to the finitely generated modules being a direct sum of uniserial modules [8]. Results along this line have been generalized to more arbitrary rings [6], [13].This article is concerned with investigating rings whose indecomposable injective modules are uniserial. The following question is considered which was first posed in [4]. If an artinian ring R has all indecomposable injective modules uniserial, does this imply that R is serial? The answer is yes if R is a finite dimensional algebra over a field. In this paper it is shown, provided R modulo its radical is commutative, that R has every left indecomposable injective uniserial implies that R is right serial.

scholarly journals On the group ring of a finite abelian group

1969 ◽  
Vol 1 (2) ◽  
pp. 245-261 ◽  
Author(s):  
Raymond G. Ayoub ◽  
Christine Ayoub
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Group Ring ◽  
Free Abelian Group ◽  
Finite Abelian Group ◽  
Rational Numbers ◽  
Cyclotomic Fields ◽  
Group Of Units ◽  
Direct Sum ◽  
A Finite Group

The group ring of a finite abelian group G over the field of rational numbers Q and over the rational integers Z is studied. A new proof of the fact that the group ring QG is a direct sum of cyclotomic fields is given – without use of the Maschke and Wedderburn theorems; it is shown that the projections of QG onto these fields are determined by the inequivalent characters of G. It is proved that the group of units of ZG is a direct product of a finite group and a free abelian group F and the rank of F is determined. A formula for the orthogonal idempotents of QG is found.

A CONSTRUCTION OF RIGHT ARTINIAN RIGHT SERIAL RINGS

2013 ◽  
Vol 12 (08) ◽  
pp. 1350057
Author(s):  
SURJEET SINGH
Keyword(s):  
Homomorphic Image ◽  
Artinian Ring ◽  
Finitely Generated ◽  
Serial Ring ◽  
Artinian Rings ◽  
Direct Sum ◽  
Non Local ◽  
Universal Construction ◽  
Generating System

A ring R is said to be right serial, if it is a direct sum of right ideals which are uniserial. A ring that is right serial need not be left serial. Right artinian, right serial ring naturally arise in the study of artinian rings satisfying certain conditions. For example, if an artinian ring R is such that all finitely generated indecomposable right R-modules are uniform or all finitely generated indecomposable left R-modules are local, then R is right serial. Such rings have been studied by many authors including Ivanov, Singh and Bleehed, and Tachikawa. In this paper, a universal construction of a class of indecomposable, non-local, basic, right artinian, right serial rings is given. The construction depends on a right artinian, right serial ring generating system X, which gives rise to a tensor ring T(L). It is proved that any basic right artinian, right serial ring is a homomorphic image of one such T(L).

scholarly journals The identity of certain representation algebra decompositions

1970 ◽  
Vol 3 (1) ◽  
pp. 73-74
Author(s):  
S. B. Conlon ◽  
W. D. Wallis
Keyword(s):  
Finite Group ◽  
Tensor Product ◽  
Linear Algebra ◽  
Residue Field ◽  
Complex Field ◽  
Direct Sums ◽  
Isomorphism Classes ◽  
Direct Sum ◽  
A Finite Group ◽  
Representation Algebra

Let G be a finite group and F a complete local noetherian commutative ring with residue field of characteristic p # 0. Let A(G) denote the representation algebra of G with respect to F. This is a linear algebra over the complex field whose basis elements are the isomorphism-classes of indecomposable finitely generated FG-representation modules, with addition and multiplication induced by direct sum and tensor product respectively. The two authors have separately found decompositions of A(G) as direct sums of subalgebras. In this note we show that the decompositions in one case have a common refinement given in the other's paper.

scholarly journals A Gelfand Model for Weyl Groups of Type D2n

ISRN Algebra ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-16
Author(s):  
José O. Araujo ◽  
Luis C. Maiarú ◽  
Mauro Natale
Keyword(s):  
Finite Group ◽  
Weyl Algebra ◽  
Polynomial Model ◽  
Irreducible Representations ◽  
Weyl Groups ◽  
Complex Numbers ◽  
Irreducible Factor ◽  
Direct Sum ◽  
Finite Dimensional ◽  
A Finite Group

A Gelfand model for a finite group G is a complex representation of G, which is isomorphic to the direct sum of all irreducible representations of G. When G is isomorphic to a subgroup of GLn(ℂ), where ℂ is the field of complex numbers, it has been proved that each G-module over ℂ is isomorphic to a G-submodule in the polynomial ring ℂ[x1,…,xn], and taking the space of zeros of certain G-invariant operators in the Weyl algebra, a finite-dimensional G-space 𝒩G in ℂ[x1,…,xn] can be obtained, which contains all the simple G-modules over ℂ. This type of representation has been named polynomial model. It has been proved that when G is a Coxeter group, the polynomial model is a Gelfand model for G if, and only if, G has not an irreducible factor of type D2n, E7, or E8. This paper presents a model of Gelfand for a Weyl group of type D2n whose construction is based on the same principles as the polynomial model.

scholarly journals Some characterizations of hereditarily artinian rings

1986 ◽  
Vol 28 (1) ◽  
pp. 21-23 ◽  
Author(s):  
Dinh van Huynh
Keyword(s):  
Artinian Ring ◽  
Chain Condition ◽  
Finite Ring ◽  
Artinian Rings ◽  
Direct Sum ◽  
Descending Chain Condition ◽  
Associative Rings

Throughout this note, rings will mean associative rings with identity and all modules are unital. A ring R is called right artinian if R satisfies the descending chain condition for right ideals. It is known that not every ideal of a right artinian ring is right artinian as a ring, in general. However, if every ideal of a right artinian ring R is right artinian then R is called hereditarily artinian. The structure of hereditarily artinian rings was described completely by Kertész and Widiger [5] from which, in the case of rings with identity, we get:A ring R is hereditarily artinian if and only if R is a direct sum S ⊕ F of a semiprime right artinian ring S and a finite ring F.

scholarly journals A decomposition of equivariant K-theory in twisted equivariant K-theories

2017 ◽  
Vol 28 (02) ◽  
pp. 1750016 ◽  
Author(s):  
José Manuel Gómez ◽  
Bernardo Uribe
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Irreducible Representation ◽  
Isomorphism Class ◽  
Irreducible Representations ◽  
Direct Sum ◽  
K Theory ◽  
Group 2 ◽  
A Finite Group

For [Formula: see text] a finite group and [Formula: see text] a [Formula: see text]-space on which a normal subgroup [Formula: see text] acts trivially, we show that the [Formula: see text]-equivariant [Formula: see text]-theory of [Formula: see text] decomposes as a direct sum of twisted equivariant [Formula: see text]-theories of [Formula: see text] parametrized by the orbits of the conjugation action of [Formula: see text] on the irreducible representations of [Formula: see text]. The twists are group 2-cocycles which encode the obstruction of lifting an irreducible representation of [Formula: see text] to the subgroup of [Formula: see text] which fixes the isomorphism class of the irreducible representation.

Semi-Simple Artinian Rings of Fixed Points

1975 ◽  
Vol 18 (2) ◽  
pp. 189-190 ◽  
Author(s):  
Miriam Cohen ◽  
Susan Montgomery
Keyword(s):  
Finite Group ◽  
Fixed Points ◽  
Group Of Automorphisms ◽  
Artinian Rings ◽  
A Finite Group

Let G be a finite group of automorphisms of the ring R, and let RG denote the ring of fixed points of G in R; that is, RG={x∊R|Xg = x,∀∊ G}. Let |G| denote the order of G. In this note, we prove the following:Theorem.Assume that R has no nilpotent ideals and no |G|-torsion. Then if RG is semi-simple Artinian, R is semi-simple Artinian.

Some computations in the modular representation ring of a finite group

1971 ◽  
Vol 69 (1) ◽  
pp. 163-166 ◽  
Author(s):  
John Santa Pietro
Keyword(s):  
Finite Group ◽  
Tensor Product ◽  
Modular Representation ◽  
Multiplicative Structure ◽  
Characteristic P ◽  
Isomorphism Classes ◽  
Direct Sum ◽  
Representation Ring ◽  
Ring Isomorphism ◽  
A Finite Group

Let p be an odd prime and G = HB be a semi-direct product where H is a cyclic, p-Sylow subgroup and B is finite Abelian. If K is a field of characteristic p the isomorphism classes of KG-modules relative to direct sum and tensor product generate a ring a(G) called the representation ring of G over K. If K is algebraically closed it is shown in (4) that there is a ring isomorphism a(G) ≃ a(HB2)⊗a(B1) where B1 is the kernel of the action of B on H and B2 = B/B1.> 2, Aut (H) is cyclic thus HB2 is metacyclic. The study of the multiplicative structure of a(G) is thus reduced to that of the known rings a(B1) and a(HB2) (see (3)).

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