Group rings

Centrally Essential Semirings

“…This process will terminate after a finite number of steps, i.e., there exists a positive integer k such that x k •∆(Q 8 , Z(Q 8 )) = 0 (we assume that that x 0 = x). Then x k ∈ RQ 8 Z(Q 8 ) ; e.g., see [21,Proposition 1.12]. Therefore,…”

“…Proof. If charR = 0 and R is a domain, then RG does not have nil-ideals; in particular, it is semiprime, see[21, Theorem 5.1]. It follows from[13, …”

On submodules of modules over group rings

“…The history of group rings dates back to long time and since then, many survey articles have appeared ( [3], [6], [12], [13], [17]). But modules over group rings are one of the subjects studied in recent years by a lot of authors interested in algebra ( [4], [5], [7], [8], [16]).…”

Infinite independent systems of the identities of the associative algebra over an infinite field of characteristic p > 0

“…Lemma 2 [7]. Let H be a subgroup of the group G. Then (1) if the elements h i generate the subgroup H, then the elements 1 − h i generate the right ideal ωH; (2) if h ∈ G, then 1 − h ∈ ωH if and only if h ∈ H; (3) H is a normal subgroup in G if and only if ωH is a two-sided ideal of the algebra ωG; (4) if H is a normal subgroup of the group G, then F (G/H) ∼ = F G/ωH;…”

“…Lemma 3 [7]. Let G be a locally finite p-group and let F be a field of characteristic p. Then the augmentation ideal ωG is locally nilpotent.…”