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On Ring Properties of Injective Hulls1)

Published online by Cambridge University Press:  20 November 2018

B. L. Osofsky
B. L. Osofsky*
Affiliation:
Rutgers, The State University of New Jersey
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Several authors have investigated "rings of quotients" of a given ring R. Johnson showed that if R has zero right singular ideal, then the injective hull of RR may be made into a right self injective,- regular (in the sense of von Neumann) ring (see [7] and [12]). In articles by Utumi [10], Findlay and Lambek [6], and Bourbaki [2], various structures which correspond to sub-modules of the injective hull of R are made into rings in a natural manner, in [8], Lambek points out that in each of these cases the rings constructed are subrings of Utumi' s maximal ring of right quotients, which is the maximal rational extension of R in its injective hull. Lambek also shows that Utumi's ring is canonically isomorphic to the bicommutator of the injective hull of RR if R has 1. It thus appears that a "natural" definition of the injective hull of RR as a ring extending module multiplication by R has been carried out only in the case that the injective hull is a rational extension of R. (See [12], [10], or [6] for various definitions of this concept.)

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 7 , Issue 3 , September 1964 , pp. 405 - 413
Copyright
Copyright © Canadian Mathematical Society 1964

References

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