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.
