Abstract
This chapter describes the Betti numbes and reflexive modules. A quasi-Frobenius (QF) ring R may be described as a ring with the property that all of its modules are reflexive or equivalently Exti(M, R) = 0 for all i ≥ 1 and all R-modules M. The chapter presents a class of commutative noetherian local rings, called BNSI rings, that are as different as possible from QF rings. It is shown that for such a ring, for any finitely generated module M, the vanishing of Exti(M, R) for any i > 2 implies that M is free. As a consequence, every finitely generated reflexive module is free. No proper principal ideal is the annihilator of any ideal (whereas in a QF ring, every ideal is an annihilator). All rings are assumed to be commutative noetherian local rings and all modules are unital and finitely generated. Every protective module is free and the maximal ideal of the ring R will be denoted by m.
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