Algebra univers. 46 (2001) 285 – 320 0002–5240/01/020285 – 36 $ 1.50 + 0.20/0 © Birkh¨ auser Verlag, Basel, 2001 Natural dualities for quasivarieties generated by a finite commutative ring David M. Clark, PaweL M. Idziak, Lousindi R. Sabourin, Csaba Szab ´ o and Ross Willard Dedicated to Viktor Aleksandrovich Gorbunov Abstract. Let R be a finite commutative ring with identity. If the Jacobson radical of R annihilates itself, then the quasivariety generated by R is dually equivalent to a category of structured Boolean spaces obtained in a natural way from R. If on the other hand the radical of R does not annihilate itself, then no such natural dual equivalence is possible. To illustrate the first result, a dual equivalence for the quasivariety generated by the ring Z p 2 , where p is prime, is given. 1. Introduction Stone’s 1936 [18] description of Boolean rings has two parts. In modern language it says that every Boolean ring is isomorphic to the ring of all clopen subsets of some Boolean space, and that the association between Boolean rings and the corresponding Boolean spaces is a dual equivalence between the quasivariety of Boolean rings and the category of Boolean spaces. The quasivariety of Boolean rings consists of all rings with identity that are embeddable into a power of the two element field F 2 . In 1968 Arens and Kaplansky [1] extended this result to the quasivariety consisting of all rings with identity that are embeddable into a power of the field F q . Taking G to be the automorphism group of F q and X to be a Boolean space continuously acted upon by G, they constructed the ring of G-stable continuous functions from X into F q . Their theorem says that • every commutative ring with identity in the quasivariety generated by F q is isomorphic to the ring of G-stable continuous functions from some Boolean G-space X into F q , and that Presented by Professors Kira Adaricheva and Wieslaw Dziobiak. Received October 1, 2000; accepted in final form December 27, 2000. 2000 Mathematics Subject Classfication: Primary 13Mxx; Secondary 18A40, 08C15. Key words and phrases: Natural dualities, commutative rings, quasivarieties. The support of Polish KBN grant 2P03A-031-09 (Idziak), the NSERC of Canada (Willard), and the Fields Institute for Research in Mathematical Sciences (Idziak, Szab ´ o, Willard) are gratefully acknowledged. 285