𝑛-Gorenstein rings

Abstract

The object of this note is to study commutative noetherian n-Gorenstein rings. The first result is: if each module satisfying Samuel’s conditions ( a i ) ({{\text {a}}_i}) for some i ≦ n i \leqq n is an ith syzygy, then the ring is n-Gorenstein. This is the converse to a theorem of Ischebeck. The next result characterizes n-Gorenstein rings in terms of commutativity of certain rings of endomorphisms. This answers a question of Vasconcelos. Finally the last result deals with embedding of finitely generated modules into finitely generated modules of finite projective dimension.