Isomorphisms between endomorphism rings of projective modules

José Luis García, Juan Jacobo Simón
1993 Glasgow Mathematical Journal  
Let R and 5 be arbitrary rings, R M and S N countably generated free modules, and let <p:End{ R M)-+End( s N) be an isomorphism between the endomorphism rings of M and N. Camillo [3] showed in 1984 that these assumptions imply that R and S are Morita equivalent rings. Indeed, as Bolla pointed out in [2] , in this case the isomorphism (p must be induced by some Morita equivalence between R and 5. The same holds true if one assumes that R M and S N are, more generally, non-finitely generated free
more » ... modules. In this note, we make the observation that the above results of Camillo and Bolla cannot be extended to a class of modules broader than that of non-finitely generated free modules in any natural way. More precisely, let M be now the class of all the countably generated locally free projective modules (over arbitrary rings); we give examples to show that: (1) there exist modules R M and S N in the class M such that End{ R M) = End( s N), while R and 5 are not Morita equivalent; (2) there exist R M in the class M and an automorphism 5 of the endomorphism ring End( R M) such that 8 cannot be induced by any Morita auto-equivalence of the ring R. All the rings in this paper are supposed to be associative and with identity element. A module R M is called locally free [4] if each finite set of elements of M is contained in a finitely generated free direct summand. If R M is a left /^-module, then End{ R M) denotes the endomorphism ring of R M (and endomorphisms act opposite scalars) and / End{ R M) will denote the subring (not necessarily with identity) of End( R M), given by / End( R M) = {f 6 End( R M)\f = g°h,h: R M-*R",g: R R"-» R M, for some integer «}. In particular, when R M is free and countably generated, then End{ R M) is isomorphic to the ring of row-finite matrices IRFM(/?) a n d / End(«A/) is then isomorphic to the subring of the matrices with a finite number of non-zero columns, FC(/?). We start with the following lemma, which will be needed for the construction of the announced examples. Notice that this lemma could also be obtained from [9, Corollary 1], but we give a different proof of it. LEMMA. Let D be a division ring. Then, the rings RFMI(D) and RFM([RFy(D)) are not Morita equivalent rings. Proof. To simplify the notation, let us put £ = [RFM(D) and 5 = RFM(£). By [1, Exercise 7, p. 23], £ has only one non-trivial ideal which is just the left socle of E, £ 0 . S has the non-trivial ideal S () = FC(£), and So-mod is a category equivalent to £-mod [5, Theorem 2.4]. By using this equivalence and [6, Proposition 3.5], we see that S" has exactly one non-trivial ideal / satisfying that S () IS () = /. However, such an / is also an ideal of 5, so that 5 has at least two non-trivial ideals. Thus, E and 5 cannot be Morita equivalent rings [1, Proposition 21.11]. t Supported by the DGICYT of Spain (PB90-0300-C02-02).
doi:10.1017/s0017089500009939 fatcat:s2pum6kn5zhevoxnb5h355ameq