scholarly journals Modules over polycyclic groups have many irreducible images

1981 ◽  
Vol 22 (2) ◽  
pp. 141-150 ◽  
Author(s):  
Kenneth A. Brown
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Noetherian Ring ◽  
Jacobson Radical ◽  
Finitely Generated ◽  
Left Module ◽  
Image Position ◽  
Polycyclic Groups ◽  
Left Annihilator ◽  
The Ideal

Recall that a Noetherian ring R is a Hilbert ring if the Jacobson radical of every factor ring of R is nilpotent. As one of the main results of [13], J. E. Roseblade proved that if J is a commutative Hilbert ring and G is a polycyclic-by-finite group then JG is a Hilbert ring. The main theorem of this paper is a generalisation of this result in the case where all the field images of J are absolute fields—we shall say that J is absolutely Hilbert. The result is stated in terms of the (Gabriel–Rentschler–) Krull dimension; the definition and basic properties of this may be found in [5]. Let M be a finitely generated right module over the ring R. We write AnnR(M) (or just Ann(M)) for the ideal {r ∈ R: Mr = 0}, the annihilator of M in R. If M is also a left module, its left annihilator will be denoted l-AnnR(M). If R is a group ring JG, put

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?

CHAIN CONDITIONS ON NON-SUMMANDS

2011 ◽  
Vol 10 (03) ◽  
pp. 475-489 ◽  
Author(s):  
PINAR AYDOĞDU ◽  
A. ÇIĞDEM ÖZCAN ◽  
PATRICK F. SMITH
Keyword(s):  
Noetherian Ring ◽  
Jacobson Radical ◽  
Finitely Generated ◽  
Chain Conditions ◽  
Finite Dimensional

Let R be a ring. Modules satisfying ascending or descending chain conditions (respectively, acc and dcc) on non-summand submodules belongs to some particular classes [Formula: see text], such as the class of all R-modules, finitely generated, finite-dimensional and cyclic modules, are considered. It is proved that a module M satisfies acc (respectively, dcc) on non-summands if and only if M is semisimple or Noetherian (respectively, Artinian). Over a right Noetherian ring R, a right R-module M satisfies acc on finitely generated non-summands if and only if M satisfies acc on non-summands; a right R-module M satisfies dcc on finitely generated non-summands if and only if M is locally Artinian. Moreover, if a ring R satisfies dcc on cyclic non-summand right ideals, then R is a semiregular ring such that the Jacobson radical J is left t-nilpotent.

scholarly journals Inertial subalgebras of algebras possessing finite automorphism groups

1979 ◽  
Vol 28 (3) ◽  
pp. 335-345 ◽  
Author(s):  
Nicholas S. Ford
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Jacobson Radical ◽  
Automorphism Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Fixed Ring ◽  
A Finite Group ◽  
Necessary And Sufficient

AbstractLet R be a commutative ring with identity, and let A be a finitely generated R-algebra with Jacobson radical N and center C. An R-inertial subalgebra of A is a R-separable subalgebra B with the property that B+N=A. Suppose A is separable over C and possesses a finite group G of R-automorphisms whose restriction to C is faithful with fixed ring R. If R is an inertial subalgebra of C, necessary and sufficient conditions for the existence of an R-inertial subalgebra of A are found when the order of G is a unit in R. Under these conditions, an R-inertial subalgebra B of A is characterized as being the fixed subring of a group of R-automorphisms of A. Moreover, A ⋍ B ⊗R C. Analogous results are obtained when C has an R-inertial subalgebra S ⊃ R.

The mod ℭ Suspension Theorem

1969 ◽  
Vol 21 ◽  
pp. 684-701 ◽  
Author(s):  
Benson Samuel Brown
Keyword(s):  
Finite Group ◽  
Exact Sequence ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Finite Abelian Groups ◽  
Cw Complexes ◽  
Image Position ◽  
A Finite Group

Our aim in this paper is to prove the general mod ℭ suspension theorem: Suppose that X and Y are CW-complexes,ℭ is a class offinite abelian groups, and that(i) πi(Y) ∈ℭfor all i < n,(ii) H*(X; Z) is finitely generated,(iii) Hi(X;Z) ∈ℭfor all i > k.Then the suspension homomorphismis a(mod ℭ) monomorphism for 2 ≦ r ≦ 2n – k – 2 (when r= 1, ker E is a finite group of order d, where Zd∈ ℭ and is a (mod ℭ) epimorphism for 2 ≦ r ≦ 2n – k – 2The proof is basically the same as the proof of the regular suspension theorem. It depends essentially on (mod ℭ) versions of the Serre exact sequence and of the Whitehead theorem.

scholarly journals On the length of the powers of systems of parameters in local ring

1990 ◽  
Vol 120 ◽  
pp. 77-88 ◽  
Author(s):  
Nguyen Tu Cuong
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Noetherian Ring ◽  
Finitely Generated ◽  
Systems Of Parameters ◽  
System Of Parameters ◽  
The Ideal

Throughout this note, A denotes a commutative local Noetherian ring with maximal ideal m and M a finitely generated A-module with dim (M) = d. Let x1, …, xd be a system of parameters (s.o.p. for short) for M and I the ideal of A generated by x1, …, xd.

On ideals preserving generalized local cohomology modules

2015 ◽  
Vol 15 (01) ◽  
pp. 1650019 ◽  
Author(s):  
Tsutomu Nakamura
Keyword(s):  
Positive Integer ◽  
Noetherian Ring ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Local Cohomology Modules ◽  
Generalized Local Cohomology Modules ◽  
Cohomology Modules ◽  
The Ideal

Let R be a commutative Noetherian ring, 𝔞 an ideal of R and M, N two finitely generated R-modules. Let t be a positive integer or ∞. We denote by Ωt the set of ideals 𝔠 such that [Formula: see text] for all i < t. First, we show that there exists the ideal 𝔟t which is the largest in Ωt and [Formula: see text]. Next, we prove that if 𝔡 is an ideal such that 𝔞 ⊆ 𝔡 ⊆ 𝔟t, then [Formula: see text] for all i < t.

The Group Ring Of a Class Of Infinite Nilpotent Groups

1955 ◽  
Vol 7 ◽  
pp. 169-187 ◽  
Author(s):  
S. A. Jennings
Keyword(s):  
Nilpotent Group ◽  
Group Ring ◽  
Characteristic Zero ◽  
Discrete Group ◽  
Nilpotent Groups ◽  
Zero Element ◽  
Free Nilpotent Group ◽  
Image Position ◽  
Torsion Free ◽  
The Ideal

Introduction. In this paper we study the (discrete) group ring Γ of a finitely generated torsion free nilpotent group over a field of characteristic zero. We show that if Δ is the ideal of Γ spanned by all elements of the form G − 1, where G ∈ , thenand the only element belonging to Δw for all w is the zero element (cf. (4.3) below).

Natural maps of extension functors and a theorem of R. G. Swan

1961 ◽  
Vol 57 (3) ◽  
pp. 489-502 ◽  
Author(s):  
P. J. Hilton ◽  
D. Rees
Keyword(s):  
Finite Group ◽  
Exact Sequence ◽  
Group Ring ◽  
Projective Module ◽  
Additive Group ◽  
Integral Group Ring ◽  
Integral Group ◽  
Natural Maps ◽  
Image Position ◽  
A Finite Group

The present paper has been inspired by a theorem of Swan(5). The theorem can be described as follows. Let G be a finite group and let Γ be its integral group ring. We shall denote by Z an infinite cyclic additive group considered as a left Γ-module by defining gm = m for all g in G and m in Z. By a Tate resolution of Z is meant an exact sequencewhere Xn is a projective module for − ∞ < n < + ∞, and.

On Group Rings

1970 ◽  
Vol 22 (2) ◽  
pp. 249-254 ◽  
Author(s):  
D. B. Coleman
Keyword(s):  
Nilpotent Group ◽  
Commutative Ring ◽  
Group Ring ◽  
Characteristic Zero ◽  
Sylow Subgroup ◽  
Jacobson Radical ◽  
Closed Field ◽  
Group Rings ◽  
Locally Finite ◽  
Image Position

Let R be a commutative ring with unity and let G be a group. The group ring RG is a free R-module having the elements of G as a basis, with multiplication induced byThe first theorem in this paper deals with idempotents in RG and improves a result of Connell. In the second section we consider the Jacobson radical of RG, and we prove a theorem about a class of algebras that includes RG when G is locally finite and R is an algebraically closed field of characteristic zero. The last theorem shows that if R is a field and G is a finite nilpotent group, then RG determines RP for every Sylow subgroup P of G, regardless of the characteristic of R.

Principally quasi-Baer properties of group rings

2012 ◽  
Vol 49 (4) ◽  
pp. 454-465
Author(s):  
Libo Zan ◽  
Jianlong Chen
Keyword(s):  
Finite Group ◽  
Left Ideal ◽  
Group Ring ◽  
Group Rings ◽  
A Finite Group ◽  
Left Annihilator

A ring R is called left p.q.-Baer if the left annihilator of a principal left ideal is generated, as a left ideal, by an idempotent. It is first proved that for a ring R and a group G, if the group ring RG is left p.q.-Baer then so is R; if in condition G is finite then |G|−1 ∈ R. Counterexamples are given to answer the question whether the group ring RG is left p.q.-Baer if R is left p.q.-Baer and G is a finite group with |G|−1 ∈ R. Further, RD∞ is left p.q.-Baer if and only if R is left p.q.-Baer.

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