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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 more or equal 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.