Artinian Band Sums of Rings

Band sums of associative rings were introduced by Weissglass in 1973. The main theorem claims that the support of every Artinian band sum of rings is finite. This result is analogous to the well-known theorem on Artinian semigroup rings. 1991 Mathematics subject classification (Amer. Math. Soc): primary 16P20, 16W50; secondary 20M25. Let B be a band, that is, a semigroup consisting of idempotents. An associative ring R is called a band sum of its subrings Rb, b e B, if R — ®6 e B Rb, the sum is direct for additive groups, and RaRb != Rab for every a, b e B. Band sums of rings were introduced in [11]. A discussion of relations between band sums and other ring and semigroup constructions is contained in [5], where several examples of effective applications of band sums are presented. Many articles have been devoted to the investigations of various properties of band sums. In particular, a wide range of properties were considered in [1]. For a fairly complete list of relevant references the reader may turn to [5]. Here we mention only the more recent articles [6, 8]. The aim of this paper is to investigate Artinian band sums of rings. Looking for a way of research it seems natural to consider the well-known results on Artinian semigroup rings as a guide. It was proved in [12] that if a semigroup ring RS is Artinian then R is Artinian and S is finite. (The proof was simplified in [7, 9].) If R and S are commutative, a complete description of Artinian RS was obtained in [4]. We shall deal with an Artinian band sum R = ®fcefi Rb and deduce analogous results. Our main theorem asserts that the set [b \ Rb & 0} is finite. Clearly, B can be infinite since there may be infinitely many zero rings among the Rb. Two examples will be given to show in general that each Rb being Artinian is neither a necessary nor sufficient condition for R to be Artinian. In the important case © 1995 Australian Mathematical Society 0263-6115/95 $A2.00 + 0.00