1964/09/01 by Barbara L. Osofsky · 2 citations
Mathematics · #Rings, Modules, and Algebras #Commutative Algebra and Its Applications #Advanced Topics in Algebra #Mathematics #Injective function #Ring (chemistry) #Hull #Quotient #Injective module #Ideal (ethics) #Pure mathematics #Maximal ideal #Multiplication (music) #Extension (predicate logic) #Zero (linguistics) #Combinatorics #Discrete mathematics
paper · pdf · doi:10.4153/cmb-1964-039-3
openalex publication_date 1964/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
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 R R 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 R R if R has 1. It thus appears that a "natural" definition of the injective hull of R R 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.)