Restricted Perfect Group Rings

2016 ◽  
Vol 44 (9) ◽  
pp. 4097-4103 ◽  
Author(s):  
Dinesh Udar ◽  
R. K. Sharma ◽  
J. B. Srivastava
Keyword(s):  
Group Rings ◽  
Perfect Group

Some Results on Semi-Perfect Group Rings

1974 ◽  
Vol 26 (1) ◽  
pp. 121-129 ◽  
Author(s):  
S. M. Woods
Keyword(s):  
Group Ring ◽  
Polynomial Ring ◽  
Strong Interaction ◽  
Sufficient Conditions ◽  
Group Rings ◽  
Commutative Case ◽  
Perfect Group ◽  
Case Examples ◽  
Complete Answer ◽  
Necessary And Sufficient

The aim of this paper is to find necessary and sufficient conditions on a group G and a ring A for the group ring AG to be semi-perfect. A complete answer is given in the commutative case, in terms of the polynomial ring A[X] (Theorem 5.8). In the general case examples are given which indicate a very strong interaction between the properties of A and those of G. Partial answers to the question are given in Theorem 3.2, Proposition 4.2 and Corollary 4.3.

On Semi-Perfect Group Rings

1969 ◽  
Vol 12 (5) ◽  
pp. 645-652 ◽  
Author(s):  
W.D. Burgess
Keyword(s):  
Group Ring ◽  
Jacobson Radical ◽  
Group Rings ◽  
Perfect Group ◽  
Completely Reducible

In what follows the notation and terminology of [7] are used and all rings are assumed to have a unity element.The purpose of this note is to give some partial answers to the question: under which conditions on a ring A and a group G is the group ring AG semi-perfect?For the convenience of the reader a few definitions and results will be reviewed. A ring R is called semi-perfect if R/RadR (Jacobson radical) is completely reducible and idempotents can be lifted modulo RadR (i.e., if x is an idempotent of R/RadR there is an idempotent e of R so that e + RadR = x).

Commutative nil-clean and $\pi$-Regular Group Rings

2019 ◽  
Vol 2019 (3) ◽  
pp. 33-39
Author(s):  
P.V. Danchev
Keyword(s):  
Group Rings ◽  
Regular Group

A note on Lie nilpotent group rings

1992 ◽  
Vol 45 (3) ◽  
pp. 503-506 ◽  
Author(s):  
R.K. Sharma ◽  
Vikas Bist
Keyword(s):  
Nilpotent Group ◽  
Group Algebra ◽  
Group Rings ◽  
Group Of Units

Let KG be the group algebra of a group G over a field K of characteristic p > 0. It is proved that the following statements are equivalent: KG is Lie nilpotent of class ≤ p, KG is strongly Lie nilpotent of class ≤ p and G′ is a central subgroup of order p. Also, if G is nilpotent and G′ is of order pn then KG is strongly Lie nilpotent of class ≤ pn and both U(KG)/ζ(U(KG)) and U(KG)′ are of exponent pn. Here U(KG) is the group of units of KG. As an application it is shown that for all n ≤ p+ 1, γn(L(KG)) = 0 if and only if γn(KG) = 0.

Subgroups induced by certain ideals of free group rings

1983 ◽  
Vol 11 (22) ◽  
pp. 2519-2525 ◽  
Author(s):  
Chander Kanta Gupta
Keyword(s):  
Free Group ◽  
Group Rings

On *-clean non-commutative group rings

2016 ◽  
Vol 15 (08) ◽  
pp. 1650150 ◽  
Author(s):  
Hongdi Huang ◽  
Yuanlin Li ◽  
Gaohua Tang
Keyword(s):  
Finite Field ◽  
Local Ring ◽  
Group Algebra ◽  
Rational Numbers ◽  
Semisimple Group ◽  
Group Algebras ◽  
Group Rings ◽  
Dihedral Groups ◽  
Commutative Local Ring ◽  
Generalized Quaternion

A ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection (∗-invariant idempotent). In this paper, we consider the group algebras of the dihedral groups [Formula: see text], and the generalized quaternion groups [Formula: see text] with standard involution ∗. For the non-semisimple group algebra case, we characterize the ∗-cleanness of [Formula: see text] with a prime [Formula: see text], and [Formula: see text] with [Formula: see text], where [Formula: see text] is a commutative local ring. For the semisimple group algebra case, we investigate when [Formula: see text] is ∗-clean, where [Formula: see text] is the field of rational numbers [Formula: see text] or a finite field [Formula: see text] and [Formula: see text] or [Formula: see text].

scholarly journals Gauss Units in Integral Group Rings

1998 ◽  
Vol 204 (2) ◽  
pp. 588-596 ◽  
Author(s):  
Olaf Neisse ◽  
Sudarshan K. Sehgal
Keyword(s):  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings
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