Two applications of maschke's theorem

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Abstract
Let R ∗ G denote a crossed product of the finite group G over the ring R and let V be an R ∗ G-module. Maschke's theorem states that if 1/∣G∣ ε R and if V is completely reducible as an R-module, then V is also completely reducible as an R ∗ G -module. In this paper, we obtain two applications of this theorem, both under the assumption that R is semiprime with no ∣G∣ -torsion. The first concerns group actions and here we show that if G acts on R and if I is an essential right ideal of the fixed ring RG , then IR is essential in Rs. This result, in turn, simplifies a number of proofs already in the literature. The second application here is a short proof of a theorem of Fisher and Montgomery which asserts that the crossed product R ∗ G is semiprime.

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