Finitely Σ-CS Property of Excellent Extensions of Rings

Abstract

For a right excellent extension S of a ring R, it is proved that R is right, finitely Σ-CS if and only if S is the same. As an application of this result, a number of examples of group rings which are finitely Σ-CS are presented. This generalizes a result of Jain, et al. [5], where it was shown that F[D_∞] is CS when F is a field of characteristic ≠ 2. It is also proved that if R is a commutative domain with 2⁻¹ ∈ R and C₂ is the cyclic group of order 2, then R[C₂] is a CS-ring.