Absolutely pure modules

A module A is shown to be absolutely pure if and only if every finite consistent system of linear equations over A has a solution in A. Noetherian, semihereditary, regular and Prüfer rings are characterized according to properties of absolutely pure modules over these rings. For example, R is Noetherian if and only if every absolutely pure i?-module is injective and semihereditary if and only if the class of absolutely pure i?-modules is closed under homomorphic images. If R is a Prüfer domain, then the absolutely pure i?-modules are the divisible modules and Ext\t(M, A)=0 whenever A is divisible and M is a countably generated torsionfree .R-module. Throughout R will denote an associative ring with identity and all modules are unital. An £-module, without further qualification, will always be a left P-module. Similarly, Noetherian and semihereditary will mean left-Noetherian and left semihereditary, respectively. A submodule A of the £-module B is said to be a pure submodule if for all right £-modules M the induced map M ® rA—^M ® rB is monic. An equivalent formulation of purity more useful for our purposes is that the induced map Homje(Af, B)—*HomR(M, B/A) be surjective for all finitelypresented £-modules M—M is finitely presented if it is the quotient of a finitely generated free £-module by a finitely generated submodule. Maddox [4] has called a module absolutely pure if it is pure in every module containing it as a submodule. As we shall see, an equally appropriate appellation for such modules would be finitely injective. Now if A is pure in B and if C is a submodule of B containing A, then it is easy to see that A is pure in C. Therefore A is absolutely pure if and only if A is pure in every injective module containing A and hence if and only if A is pure in its injective envelope. Proposition 1. An R-module A is absolutely pure if and only if Extg(Af, A) =0for all finitely presented R-modules M. Proof. Let £ be the injective envelope of A. We then have the exact sequence Received by the editors November 10, 1969. AMS 1969 subject classifications. Primary 1690; Secondary 1640, 1625.