ON FINITELY GENERATED MODULES OVER NOETHERIAN RINGS
ON FINITELY GENERATED MODULES OVER NOETHERIAN RINGS
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发表时间：
2010
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通讯作者：
J. P. JANSi
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作者：
J. P. JANSi
of A into its second dual. Following the terminology introduced by Bass [2] we shall say that A is torsionless if a is a monomorphism, reflexive if o is an isomorphism. We shall also refer to Ima as the torsionless factor of A. It is easy to see that this is also A/A0 where A0 is the intersection of kernels of the elements of A*. In this paper we plan to continue the study initiated in [5] relating some of the concepts mentioned above with the functor Ext^(A,R). Since it will only appear in this form in the present paper we shall henceforth adopt the notation E"(A) = Ext"R(A,R). Also, in this paper we shall make the standing assumptions that the ring is both left and right Noetherian and all modules under consideration are finitely generated. The reason behind the latter assumption is to insure that projective modules are reflexive and that the duals of projective modules are projective; see [2]. In §1 we relate double dual embeddings with the torsionless factors of modules A such that EX(A) = 0. The proof of this result arose out of Theorem 1.4 of [5] in which it is shown that the first dual A* is a direct summand of the "third" dual A***. In §2 we introduce the concept of D-class n and show that the dual of a module of D-class n appears as the nth kernel in a projective resolution. It is then clear that properties of the ring which are phased in terms of the sort of kernels appearing in projective resolutions can also be described in terms of modules of D-class n (and their duals). In §3, we show that, modulo a special condition, a module T„ is of D-class n if and only if E\Tn) = 0 for 1 g i ^ n 1. Under Applications, §4, we relate properties of modules of D-class n to the global dimension, left finitistic dimension and left injective dimension of the ring.
