Reflexive modules and rings with self-injective dimension two
The author considers the question: When is every finitely generated left module M over a left and right noetherian ring R with \(Ext^ i_ R(M,R)=0\) reflexive for \(i=1,2?\) As main result it is shown that if R is a left and right noetherian ring, then \(inj\dim_ RR=inj\dim R_ R\leq 2\) if and only if for a finitely generated left R-module M the following conditions are equivalent: (1) M is reflexive. (2) There is an exact sequence \(0\to M\to P_ 1\to P_ 0\) of left R-modules with the \(P_ i\) projective. (3) \(Ext^ i_ R(M,R)=0\) for \(i=1,2\). The author also obtains a result for commutative noetherian rings concerning Gorenstein rings.
scientific article; zbMATH DE number 4139960