Vol. 25, No. 2, 1968

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Homological dimensions and Macaulay rings

Gerson Louis Levin and Wolmer Vasconcelos

Vol. 25 (1968), No. 2, 315–323

This paper shows some instances where properties of a local ring are closely connected with the homological properties of a single module. Particular stress is placed on conditions implying the regularity or the Cohen-Macaulay property of the ring.

First it is proved that the regularity of the local ring R is equivalent to the finiteness of the projective or injective dimensions of a nonzero module mA, where m is the maximal ideal of R and A a finitely generated R-module. Next it is shown that over Gorenstein rings the finiteness of the projective or injective dimension are equivalent notions. Then, using some change of rings, a theorem is strengthened on embedding modules of finite length into cyclic modules over certain Macaulay rings. Finally, to mimic the equivalent statement for projective dimension, it is shown that the annihilator of a module finitely generated and having finite inactive dimension must be trivial if it does not contain a nonzero divisor.

Mathematical Subject Classification
Primary: 13.90
Received: 7 December 1966
Published: 1 May 1968
Gerson Louis Levin
Wolmer Vasconcelos
Rutgers University, New Brunswick
United States