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 ?(R) and ?(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:

1. If R is a ring of prime characteristic p, the multiplicative group R* of R is p-divisible and ?(R) = 0, then there exists a twisted group algebra R _{t 1} (G/G _p ) such that R _t G/?(R _t G) ? R _{t 1} (G/G _p ) as R-algebras;

2. If R is a ring of prime characterisitic p and R* is p-divisible, then ?(R _t G) = 0 if and only if ?(R) = 0 and G _p  = 1; and

3. 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 _{t 1} H, then R _t G ₀ ? R _{t 1} H ₀ as R-algebras.