When does \(R\) Gorenstein imply \(R^ G\) Gorenstein?
Let \(R\) be a ring, \(G\) a group of automorphisms of \(R\), and \(R^G\) the fixed subring. The paper deals with several problems in the framework of the so-called ``preserving question'' that asks, given a class of rings \(\mathcal C\), under what conditions does \(R\in{\mathcal C}\) imply \(R^G\in{\mathcal C}\). The authors study the preserving question for the class of right self-injective rings and, more generally, the relation between the injective dimensions of \(R_R\) and \(R^G_{R^G}\). In relation with this problem, it is proved that if \(G\) is a finite group, then \(\text{inj.dim }R^G_{R^G}\leq\text{inj.dim }R_R\) if \(1\in\text{tr}(R)\) and either \({_{R^G}R}\) is flat or \(I=I^GR\) for every \(G\)-invariant right ideal \(I\) of \(R\). While it is known that the answer to the preserving question for right self-injective rings is negative in general, the partial positive results hitherto known assumed the group \(G\) to be finite, which is a necessary condition for the right \(RG\)-module \(R_{RG}\) to be projective. The authors obtain some positive results assuming that \(R_{RG}\) is quasi-projective rather than projective, thus avoiding the finiteness hypothesis on \(G\). Specifically, they show that if \(R_{RG}\) and \({_{RG}R}\) are quasi-projective, every \(G\)-invariant right ideal of \(R\) has a nonzero fixed element, and \(R\) is right self-injective, then \(R^G\) is right self-injective too. The last section of the paper is devoted to the study of the preserving question for commutative Gorenstein rings when the group \(G\) is finite. The question has a negative answer in general, even assuming that the order of \(G\) is invertible in \(R\). However, the authors prove that if \(R\) is a commutative Gorenstein ring such that \(1\in\text{tr}(R)\), then \(R^G\) is Gorenstein if one of the following conditions holds: (1) \({_{R^G}}R\) is flat; (2) \(I=I^GR\) for every \(G\)-invariant ideal \(I\) of \(R\). In this situation they show how to obtain an injective resolution of \(R^G_{R^G}\) from an injective resolution of \(R_R\).
scientific article; zbMATH DE number 922322
scientific article