ON INVOLUTION RINGS

Abstract

In [28], R. Wiegandt surveyed various interesting results on the structure of involution rings. Here, we extend that article by surveying more recent results on that topic due to Birkenmeier, Groenewald, Heatherely and the author ( [4], [5], [6], [10], [11] ). The main point to declare is that the notion of one-sided ideal used to describe the structure of rings without involution is no longer efficient for involution rings since in the category of the latter rings, all homomorphisms (and also embeddings) must preserve involution. Nevertheless, a one-sided ideal closed under involution is a two-sided ideal. It turned out that the notion of ∗-biideal, which is left and right symmetric, is the most appropriate and efficient one to describe the structure of involution rings. Moreover, by imposing chain conditions on ∗-biideals, the involutive versions of well known theorems (such as Wedderburn-Artin, Goldie, Litoff-´Anh, Ayoub-Dinh Van Huynh, Rees, Hilbert Bases and Embedding Theorems) can be formulated and proved.