Vol. 72, No. 1, 1977

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Chain conditions and pure-exactness

Mangesh Bhalchandra Rege and Kalathoor Varadarajan

Vol. 72 (1977), No. 1, 223–235

Let R be a ring and M any (right) R-module. For any set I let M(I) and MI denote the direct sum and respectively the direct product of copies of M indexed by the set I. For any cardinal number r, let 𝒞r denote the class of R-modules admitting a generating set of cardinality r. In this paper we study the relationship between the pure-exactness of the sequence 0 M(I) MI MI∕M(I) 0 with respect to 𝒞r under the functor HomR and chain conditions on suitably defined families of R-modules. This study led us to the introduction of five properties Ar, A(r), Dr, D(r), and Pr for any R-module M. We also study the effect of base extension (both covariant and contravariant) of the ring R on modules having any (or some) of the above mentioned properties. Finally we obtain necessary and sufficient conditions for

0 → ⊕M  α → πM α → πM α∕⊕ M α → 0

to be pure-exact with respect to 𝒞k, where {Mα} is any family of R-modules and k any integer 1.

Mathematical Subject Classification
Primary: 16A64, 16A64
Received: 1 March 1977
Published: 1 September 1977
Mangesh Bhalchandra Rege
Kalathoor Varadarajan