A construction of rings whose injective hulls allow a ring structure

Vlastimil Dlab, Claus Michael Ringel
1973 Journal of the Australian Mathematical Society  
Communicated by M. F. Newman In her paper [3], Osofsky exhibited an example of a ring R containing 16 elements which (i) is equal to its left complete ring of quotients, (ii) is not self-injective and (iii) whose injective hull HR = H( R R) allows a ring structure extending the .R-module structure of HR. In the present note, we offer a general method of constructing such rings; in particular, given a non-trivial split Frobenius algebra A and a natural n ^ 2, a certain ring of n x n matrices
more » ... n x n matrices over A provides such an example. Here, taking for A the semi-direct extension of Z/2Z by itself and n = 2, one gets the example of Osofsky. Thus, our approach answers her question on finding a non-computational method for proving the existence of such rings. Throughout the present note, A denotes a ring with unity 1. Given an Amodule M, denote by Rad M the intersection of all maximal submodules of M. Dually, if M has minimal submodules, SocM denotes their union. Also, write Top M =• M /Rad M. The radical Rad A of the ring A will be denoted consistently by W and the factor A /Wby Q. By a split ring A we shall understand a ring which is a semi-direct extension (Q, W) of W by Q; in this case, we shall consider Q to be embedded as a subring in A. Thus A = Q® W as additive groups and (^. w j (#2> W 2) = (<7i42>(h w 2 + w i<i2 + w t w 2 ). For example, it is well-known that every finite dimensional algebra over an algebraically closed field is a split ring. We recall that a Frobenius algebra A is a finite dimensional algebra over a field F which is self-injective; and that, given a decomposition A = ©f = 1 Ae i into indecomposable left ideals, there exists a permutation n of {1,2,•••,s} such that Soc Ae^ Top Ae n ( iy Given a ring R and an .R-module M, the injective hull of M will be denoted by HM, the injective hull of R R by HR. The double centralizer of HR is called the This research has been supported by the National Research Council of Canada. 7 use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700013860 Downloaded from https://www.cambridge.org/core. IP address: 207.241.231.82, on 24 Jul 2018 at 02:26:37, subject to the Cambridge Core terms of EXAMPLE. The split extension B = (Z,Z) of 1 by itself is a left order in •4 = ( Q , Q ) which satisfies the assumptions of Corollary.
doi:10.1017/s1446788700013860 fatcat:6h55qqirffeoxctbx7zd3v4mpy