Vol. 28, No. 3, 1969

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Purity and algebraic compactness for modules

Robert Breckenridge Warfield, Jr.

Vol. 28 (1969), No. 3, 699–719

A submodule A of a left module B (over an associative ring with 1) is pure if for any right module F, the natural homomorphism F A F B is injective. A module C is pure-injective if for any module B and pure submodule A, any homomorphism from A to C extends to B. The theory of this notion of purity and the corresponding class of pure-injectives is developed in this paper, with special attention to modules over commutative Noetherian rings and Prüfer rings. It is proved that pure-injective envelopes exist and the pure-injective modules are characterized as retracts of topologically compact modules. For this reason, the pure-injective modules are also called algebraically compact. For modules over Prüfer rings, certain simplifications occur, due essentially to the fact that a finitely presented module is a summand of a direct sum of cyclic modules. Complete sets of invariants are obtained for certain classes of algebraically compact modules over certain Prüfer rings.

Mathematical Subject Classification
Primary: 16.40
Received: 28 March 1968
Published: 1 March 1969
Robert Breckenridge Warfield, Jr.