On the Commutative Twisted Group Algebras

Abstract

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:

1. 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 _{t 1} (G/G _p ) such that R _t G/ℬ(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 ℬ(R _t G) = 0 if and only if ℬ(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.

Key Words:

• Commutative twisted group algebras
• Isomorphism

AMS Mathematics Subject Classification (2000):

• 16S35
• 13A10
• 16N40
• 16S34

ACKNOWLEDGMENTS

Research was partially supported by the fund “NI” of Plovdiv University, Bulgaria.

Notes

Communicated by V. A. Artamonov.