A construction of rings whose injective hulls allow a ring structure

Dlab V, Ringel CM (1973)
The journal of the Australian Mathematical Society 16(01): 7-13.

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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.
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Dlab V, Ringel CM. A construction of rings whose injective hulls allow a ring structure. The journal of the Australian Mathematical Society. 1973;16(01):7-13.
Dlab, V., & Ringel, C. M. (1973). A construction of rings whose injective hulls allow a ring structure. The journal of the Australian Mathematical Society, 16(01), 7-13.
Dlab, V., and Ringel, C. M. (1973). A construction of rings whose injective hulls allow a ring structure. The journal of the Australian Mathematical Society 16, 7-13.
Dlab, V., & Ringel, C.M., 1973. A construction of rings whose injective hulls allow a ring structure. The journal of the Australian Mathematical Society, 16(01), p 7-13.
V. Dlab and C.M. Ringel, “A construction of rings whose injective hulls allow a ring structure”, The journal of the Australian Mathematical Society, vol. 16, 1973, pp. 7-13.
Dlab, V., Ringel, C.M.: A construction of rings whose injective hulls allow a ring structure. The journal of the Australian Mathematical Society. 16, 7-13 (1973).
Dlab, Vlastimil, and Ringel, Claus Michael. “A construction of rings whose injective hulls allow a ring structure”. The journal of the Australian Mathematical Society 16.01 (1973): 7-13.
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