On the commutative twisted group algebras

Let G be an abelian group and let R be a commutative ring with identity. Denote by R(t)G a commutative twisted group algebra (a commutative twisted group ring) of G over R, by B(R) and B(R(t)G) the nil radicals of R and R(t)G, respectively, by G(p) the p-component of G and by G(0) the torsion subgroup of G. We prove that: (i) If R is a ring of prime characteristic p, the multiplicative group R* of R is p-divisible and B(R) = 0, then there exists a twisted group algebra R-t1(G/G(p)) such that R(t)G/B(R(t)G)congruent to R-t1(G/Gp) as R-algebras; (ii) If R is a ring of prime characterisitic p and R* is p-divisible, then B(R(t)G)= 0 if and only if B(R)= 0 and G(p) = 1; and (iii) If B(R) = 0, the orders of the elements of G(0) are not zero divisors in R, H is any group and the commutative twisted group algebra R(t)G is isomorphic as R-algebra to some twisted group algebra R-t1 H, then R(t)G(0) congruent to Rt1H0 as R-algebras.