a note on central reversible group ring

In this article, we study conditions under which a group ring r[g] of a group g over a ring r becomes a central reversible ring. the notion of central reversible ring is a generalization of reversible ring. we see that the group ring k[g] of a group g over a field k of characteristic zero is reversible if and only if k[g] is central reversible if and only if k[g] is reduced. also, if k is a field of characteristic p > 0 and g is a group which has no finite normal subgroup of order divisible by p, then k[g] is reversible if and only if k[g] is central reversible. finally, we show that the group ring k[q8] of quaternions q8 over a field k of characteristic zero is reversible if and only if k[q8] is central reversible if and only if the equation 1 + x2 + y2 = 0 is not solvable in k.