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A construction of rings whose injective hulls allow a ring structure
Published online by Cambridge University Press: 09 April 2009
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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(RR) 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 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.
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- Research Article
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- Copyright © Australian Mathematical Society 1973
References
[1]Curtis, C. W. and Reiner, I., Representation theory of finite groups and associative algebras Interscience (New York: Inc., 1962).Google Scholar
[3]Osofsky, B., ‘On ring properties of injective hulls’, Canad. Math. Bull. 7 (1964), 405–412.Google Scholar
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