scholarly journals Rank Element of a Projective Module

1965 ◽  
Vol 25 ◽  
pp. 113-120 ◽  
Author(s):  
Akira Hattori
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Local Ring ◽  
Projective Module ◽  
Projective Modules ◽  
Local Theorem ◽  
Complete Local Ring ◽  
A Finite Group

In § 1 of this note we first define the trace of an endomorphism of a projective module P over a non-commutative ring A. Then we call the trace of the identity the rank element r(P) of P, which we shall illustrate by several examples. For a projective module P over the groupalgebra of a finite group G, the rank element of P is essentially the character of G in P. In § 2 we prove that under certain assumption two projective modules Pi and P2 over an algebra over a complete local ring o are isomorphic if and only if their rank elements are identical. This is a type of proposition asserting that two representations are equivalent if and only if their characters are identical, and in fact, when A is the groupalgebra, the above theorem may be considered as another formulation of Swan’s local theorem [9]).

scholarly journals Adams operations for projective modules over group rings

1997 ◽  
Vol 122 (1) ◽  
pp. 55-71 ◽  
Author(s):  
BERNHARD KÖCK
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Grothendieck Group ◽  
Base Change ◽  
Group Rings ◽  
Projective Modules ◽  
Power Operations ◽  
Definition Of ◽  
A Finite Group ◽  
Twisted Group Ring

Let R be a commutative ring, Γ a finite group acting on R, and let k∈ℕ be invertible in R. Generalizing a definition of Kervaire, we construct an Adams operation ψk on the Grothendieck group and on the higher K-theory of projective modules over the twisted group ring R#Γ. For this, we generalize Atiyah's cyclic power operations and use shuffle products in higher K-theory. For the Grothendieck group, we show that ψk is multiplicative and that it commutes with base change, with the Cartan homomorphism, and with ψl for any other l which is invertible in R.

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.

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.

scholarly journals Actions of finite groups on commutative rings I

2003 ◽  
Vol 34 (3) ◽  
pp. 201-212
Author(s):  
A. G. Naoum ◽  
W. K. Al-Aubaidy
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Finite Groups ◽  
Commutative Rings ◽  
Group Of Automorphisms ◽  
A Finite Group ◽  
The Ideal

Let $R$ be a commutative ring with 1, and let $G$ be a finite group of automorphisms of $R$. Denote by $R^G$ the fixed subring of $G$, and let $I$ be a subset of $R^G$. In this paper we prove that if the ideal generated by $I$ in $R$ satisfies a certain property with regard to projectivity, flatness, multiplication or related concepts, then the ideal generated by $I$ in $R^G$ also satisfies the same property.

scholarly journals On modules over groups

Filomat ◽  
2016 ◽  
Vol 30 (4) ◽  
pp. 1021-1027 ◽  
Author(s):  
Mehmet Uc ◽  
Ortac Ones ◽  
Mustafa Alkan
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Endomorphism Ring ◽  
A Finite Group

For a finite group G, by the endomorphism ring of a module M over a commutative ring R, we define a structure for M to make it an RG-module so that we study the relations between the properties of R-modules and RG-modules. Mainly, we prove that RadRM is an RG-submodule of M if M is an RG- module; also RadRM ? RadRGM where RadAM is the intersection of the maximal A-submodule of module M over a ring A. We also verify that M is an injective (projective) R-module if and only if M is an injective (projective) RG-module.

Varieties and modules with vanishing cohomology

1994 ◽  
Vol 116 (2) ◽  
pp. 245-251 ◽  
Author(s):  
Jon F. Carlson ◽  
Geoffrey R. Robinson
Keyword(s):  
Finite Group ◽  
Projective Module ◽  
Group Algebras ◽  
Finite Characteristic ◽  
Principal Block ◽  
Vanishing Of Cohomology ◽  
A Finite Group ◽  
Special Case

Several years ago the authors, together with Dave Benson, conducted an investigation into the vanishing of cohomology for modules over group algebras [2]. It was mostly in the context of kG-modules where k is a field of finite characteristic p and G is a finite group whose order is divisible by p. Aside from some general considerations, the main results of [2] related the existence of kG-modules M with H*(G, M) = 0 to the structure of the centralizers of the p-elements in G. Specifically it was shown that there exists a non-projective module M in the principal block of kG with H*(G, M) = 0 whenever the centralizer of some p-element of G is not p-nilpotent. The converse was proved in the special case that the prime p is an odd integer (p > 2). In addition there was some suspicion and much speculation about the structure of the varieties of such modules. However, proofs seemed to be waiting for a new idea.

Homological Dimensions of Skew Group Rings

2020 ◽  
Vol 27 (02) ◽  
pp. 319-330
Author(s):  
Yueming Xiang
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Group Ring ◽  
Global Dimension ◽  
Group Rings ◽  
Separable Extension ◽  
Homological Dimensions ◽  
A Finite Group ◽  
Coefficient Ring ◽  
Skew Group Ring

Let R be a ring and let H be a subgroup of a finite group G. We consider the weak global dimension, cotorsion dimension and weak Gorenstein global dimension of the skew group ring RσG and its coefficient ring R. Under the assumption that RσG is a separable extension over RσH, it is shown that RσG and RσH share the same homological dimensions. Several known results are then obtained as corollaries. Moreover, we investigate the relationships between the homological dimensions of RσG and the homological dimensions of a commutative ring R, using the trivial RσG-module.

On a Class of Projective Modules Over Central Separable Algebras

1971 ◽  
Vol 14 (3) ◽  
pp. 415-417 ◽  
Author(s):  
George Szeto
Keyword(s):  
Finite Number ◽  
Commutative Ring ◽  
Projective Module ◽  
Projective Modules ◽  
Finitely Generated ◽  
Perfect Field ◽  
Maximal Ideals ◽  
Ring Of Polynomials ◽  
Separable Algebras

In [5], DeMeyer extended one consequence of Wedderburn's theorem; that is, if R is a commutative ring with a finite number of maximal ideals (semi-local) and with no idempotents except 0 and 1 or if R is the ring of polynomials in one variable over a perfect field, then there is a unique (up to isomorphism) indecomposable finitely generated projective module over a central separable R-algebra A.

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