Certain classes of ideals in group rings

Abstract

Let R be a ring with identity and H a normal subgroup of the group G. In this paper the relationship between certain classes of ideals in RH (or R) and the group ring RG is investigated by employing McCoy's “going up” and “going down” method which he used for polynomial rings in [2]. From the results obtained it is inferred that P(R)S = P(RS) if S is an u.p. - semigroup with unity, wher P(R) is the prime radical of R. If H is a central subgroup of G such that G/H is an u.p.- group, then P(RG) = P(RH)RG. Furthermore, if L(R) denotes the Levitzki nil radical of R, then it is proved that if R is any ring and S and ordered semigroup with unity, then L(RS) = L(R)S. Also, if G/H can be ordered then L(RH) ∗ RG = L(RG), while L(RH) = L(RG) ∩ RH for any central subgroup H of G. If the upper nil radical of R is denoted by U(R), and H is normal subgroup of G such that G/H can be ordered, then U(RG) ⊆ U(RH) ∗RG. If R is any ring and S an ordered semigroup with unity, then we have U(RS) ⊆ U(R)S.