Vol. 35, No. 1, 1970

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Torsion-free and divisible modules over matrix rings

David R. Stone

Vol. 35 (1970), No. 1, 235–253

A short exact sequence 0 K F E 0 of left modules over a ring A is l-pure if aK = K aF for all a A, and pure if for any right A-module M, the map M K M F is injective. A module E is torsion free (Iiattori) if its presence on the right forces l-purity, and flat if it forces purity. Similarly, we have on the left the notions of divisibility (Hattori) and absolute purity. Considering the functor E En taking A-modules to modules over the matrix ring Mn(A), a sequence is called n-pure if its image under this functor is 1-pure; n-torsion free and n-divisible modules are similarly defined. It is shown that purity, flatness, and absolute purity, respectively, are equivalent to the requirement that n-purity, n-torsion-freeness, and n-divisibility should hold for all n. n-divisibility and absolute purity are preserved under direct sums, products and certain inductive limits; n-torsion-freeness and flatness under direct sums and inductive limits, but not products. A condition is given guaranteeing that products of at most a given cardinality preserve n-torsion-freeness. It is shown that if every left ideal of A is generated by at most n elements, then n-torsion-freeness is equivalent to flatness. The behavior of these properties under localization is studied, and it is shown that if A is locally a domain then the two notions of purity agree if and only if w. gl. dim. (A) 1.

Mathematical Subject Classification
Primary: 16.40
Received: 10 September 1968
Revised: 4 March 1970
Published: 1 October 1970
David R. Stone