Let KG be the group ring of a group G over a commutative ring K with unity. The rings KG are described for which xxσ = xσx for all x = ∑ g∈G αgg ∈ KG, where x → xσ = ∑ g∈G αgf(g)σ(g) is an involution of KG; here f : G → U(K) is a homomorphism and σ is an antiautomorphism of order two of G. Let R be a ring with unity. We denote by U(R) the group of units of R. A (bijective) map : R → R is called...

A code over a group ring is defined to be a submodule of that group ring. For a code C over a group ring RG, C is said to be checkable if there is v ∈ RG such that C = {x ∈ RG : xv = 0}. In [6], Jitman et al. introduced the notion of code-checkable group ring. We say that a group ring RG is code-checkable if every ideal in RG is a checkable code. In their paper, Jitman et al. gave a necessary a...

Group Rings Let G = g1, g2, . . . , gn be a finite group, and let k be a field. We define the group ring k[G] to be the set of sums of the form a1g1 + a2g2 + · · ·+ angn with each ai ∈ k and gi ∈ G. Addition is defined componentwise, i.e. (a1g1 + a2g2 + · · ·+ angn) + (b1g1 + b2g2 + · · ·+ bngn) = ((a1 + b1)g1 + (a2 + b2)g2 + · · ·+ (an + bn)gn). We define multiplication in the following way: (...

In this paper we introduce a Group Key Management protocol following the idea of classical that extends well-known Diffie–Hellman key agreement to group users. The is defined in non-commutative setting, more precisely, twisted dihedral ring. for an arbitrary cocycle, extending previous agreements considered two main objective work show there no lack security derived from fact larger amount publ...

Let G be a multiplicative group, K a commutative ring with unit, and K(G) the group ring of G with respect to K. We say that K(G) is regular if given an x in K(G), there is a y in K(G) such that xyx = x. Using a homological characterization of regular rings which was found independently by M. Harada [2, Theorem 5] and the author, we prove that if G is locally finite, then K(G) is regular if and...

Since the foundation paper of M. Auslander and 0. Goldman [ 1 J, several authors have generalized constructions for studying group algebras over fields to separable algebras over commutative rings. Specifically, we have in mind the Schur index [20], the Schur exponent [18], the Schur group [S], and the uniform group [ 111. This paper gives additional properties of these constructions and studie...