Reversible Rings

Commutative rings form a very special subclass of rings, which shows quite different behaviour from the general case. For example, in a (non‐trivial) commutative ring, the absence of zero‐divisors is sufficient as well as necessary for the existence of a field of fractions, whereas for general rings, another infinite set of conditions is needed to characterize subrings of skew fields. This suggests the study of a class of rings which includes all commutative rings as well as all integral domains: reversible rings, where a ring is called reversible if ab = 0 implies ba = 0. It turns out that this condition helps to simplify other ring conditions, as we shall see in Section 2, although most of these results are at a somewhat superficial level. We therefore introduce a more technical notion, full reversibility, in Section 3, and show that this is the precise condition for the least matrix ideal to be proper and consist entirely of non‐full matrices. Further, we show in Section 4 that a fully reversible ring is embeddable in a skew field if and only if it is an integral domain. In what follows, all rings are associative, with a unit element 1 which is preserved by ring homomorphisms, inherited by subrings and acts unitally on modules. I am grateful to V. de O. Ferreira for his comments, in particular the suggestion of using the notion of unit‐stable rings. 1991 Mathematics Subject Classification 16U80.