Note on Morita equivalence in ring extensions

TL;DR

This paper explores Morita invariance in various classes of ring extensions, demonstrating which are invariant and providing an example of a non-invariant class, thus clarifying the role of Morita equivalence in ring theory.

Contribution

It proves Morita invariance for several classes of ring extensions and presents an example of a class that is not Morita invariant.

Findings

Morita invariance holds for multiple classes of ring extensions.

Examples of classes that are not Morita invariant are provided.

The results clarify the significance of Morita equivalence in classifying ring extensions.

Abstract

It seems that Morita invariance is a useful criterion for judging the importance of the classes of ring extensions concerned. Y. Miyashita introduced the notion of Morita equivalence in ring extensions, and he showed that the classes of $G$-Galois extensions and Frobenius extensions are Morita invariant. After that, S. Ikehata showed that the classes of separable extensions, Hirata separable extensions, symmetric extensions, and QF-extensions are Morita invariant. In this paper, we shall prove that the classes of several extensions are Morita invariant. Further, we will give an example of the class of ring extensions which is not Morita invariant.