Biregular crossed products (Q1103031)

A ring R is said to be biregular if every principal two-sided ideal of R is generated by a central idempotent. This paper studies biregular crossed products K*G. Suppose K is a biregular ring and G is a finite group with \(| G|^{-1}\in K\). The main result here asserts that if either K satisfies a.c.c. or d.c.c. on principal two sided ideals, or if K is commutative and K*G is a twisted group ring, then K*G is biregular. Author's remark. (April 1988). The proof of Lemma 1 is valid only if R is a twisted group ring (i.e. \(\sigma =1)\) over a commutative ring K. Here we choose the element \(c\in L\) such that \(\pi (c)=e\) and \(ce=c\). Otherwise, we set \(L'=\{m\in L|\pi (m)=0\}\). When the biregular ring K satisfies the chain condition (d.c.c. or a.c.c.) on principal two-sided ideals, then the (K,K)-module M satisfies the maximum condition on (K,K)- submodules, since K is a finite sum of simple rings [\textit{V. A. Andrunakievich}, Mat. Sb., Nov. Ser. 39(81), 447-464 (1956; Zbl 0071.029)]. It follows that L' is a finitely generated (K,K)-submodule and the proof also holds. Therefore, Theorems 3 and 6 and Corollary 7 are valid if K satisfies a chain condition on principal two-sided ideals or if R is a twisted group ring over a commutative ring. Reviewer's remark. It is a simple matter to construct a counterexample to Theorem 3 as originally stated in the paper. Let F be a field of any characteristic and let K be the subring of \(\prod^{\infty}_{i=1}F\) consisting of all sequences which are eventually constant. Then K is a commutative von Neumann regular ring and the group \(G=\{1,g\}\) of order 2 acts on K by interchanging the 2i-1 and 2i coordinates for all \(i\geq 1\). Form the skew group ring \(R=KG\). Since the annihilator in K of \(\{k^ g- k|\) \(k\in K\}\) is zero, it follows that the center and hence all central idempotents of R are contained in K. Thus, since R has F[G] as a homomorphic image, we see that the ideal R(1-g)R is proper and is not generated by a central idempotent.