Semigroup methods in ring theory

Abstract

Let A be a ring and denote by R(A) the Jacobson radical. We will write R instead of R(A) if it is clear from the context in which ring we operate. The question of lifting idempotents of the ring A/R to the ring A arises in various contexts in ring theory. However, what one would frequently like to know beyond the existence of idempotents in A mapping onto a given set of idempotents in A/R is whether or not a given set of orthogonal (summable) idempotents of A/R can be lifted orthogonally (summably) to d. In general these questions have a negative answer. There arc rings which contain idempotents modulo the radical which cannot be lifted. There are rings containing orthogonal idempotents modulo the radical which cannot be lifted orthogonally, inspite of the fact that each single idempotent can be lifted. For an example see ([13], 3.A). It will be shown, however, that any countable set of orthogonal idempotents of A/R can be lifted orthogonally to A provided the individual idempotents can be lifted. The approach to the indicated problem will be semigroup theoretical. Extensive use will be made of the structure theory of completely simple semigroups ([5], pp. 76). This procedure has several advantages. Firstly, it leads to more general results, even in very simple cases [compare e.g. ([7], Prop. 5, p. 54) with (Corollary 18). K o reference to the actual lifting is needed]. Secondly, the results are obtained in a more conceptual way. Almost all of the often very tricky calculations can be avoided (compare e.g. [8], Lemma 12, p. 166 and (22) or [13], 4.6 and (21)). Unfortunately our method will not be very helpful in the lifting of single idempotents. It turns out that the lifting of an idempotent is tantamount to the existence of a completely simple minimal ideal in a certain semigroup. In general, however, this is an equally inaccessible problem. On the contrary