If ,
...,
are indeterminates over a field , the above condition is fulfilled by the maximal ideal . In fact, the notion
of Cohen-Macaulay ring was inspired by polynomial rings. The above property was proven
for the first time by Macaulay (1916) for every ideal in a polynomial ring over the
complex field.
The class of Cohen-Macaulay rings contains the class of Gorenstein rings, which includes all regular local rings (Bruns and Herzog 1998, p. 95).
These were intensively studied by Cohen (1946, pp. 85-106).
Balcerzyk, S. and Józefiak, T. "Cohen-Macaulay Rings." Ch. 3 in Commutative
Rings: Dimension, Multiplicity and Homological Methods. Chichester, England:
Ellis Horwood, pp. 101-107, 1989.Bourbaki, N. "Anneaux de
Macaulay." §2.5 in Eléments de mathématiques, Chap. 10,
Algèbre Commutative. Paris, France: Masson, pp. 30-32, 1998.Bruns,
W. and Herzog, J. Cohen-Macaulay
Rings, 2nd ed. Cambridge, England: Cambridge University Press, 1998.Cohen,
I. S. "On the Structure and Ideal Theory of Complete Local Rings."
Trans. Amer. Math. Soc.59, 54-106, 1946.Hutchins, H. H.
Examples
of Commutative Rings. Passaic, NJ: Polygonal Publishing House, 1981.Kunz,
E. "Regular Sequences, Cohen-Macaulay Rings and Modules." §6.3 in
Introduction
to Commutative Algebra and Algebraic Geometry. Boston, MA: Birkhäuser,
pp. 183-191, 1985.Macaulay, F. S. The
Algebraic Theory of Modular Systems. Cambridge, England: Cambridge University
Press, 1916.Matsumura, H. "Cohen-Macaulay rings." §17
in Commutative
Ring Theory. Cambridge, England: Cambridge University Press, pp. 133-139,
1986.Samuel, P. and Zariski, O. "Macaulay rings." §A6
in Commutative
Algebra, Vol. 2. Princeton, NJ: Van Nostrand, pp. 394-403, 1958.Sharp,
R. Y. "Cohen-Macaulay Rings." Ch. 17 in Steps
in Commutative Algebra, 2nd ed. Cambridge, England: Cambridge University
Press, pp. 330-344, 2000.
in which any proper ideal 
of height 
contains a sequence 
, ..., 
of elements (called a ring
regular sequence such that for all 
, ..., 
, the residue class of 
in the quotient
ring 
is a non-zero divisor.
,
..., 
are indeterminates over a field 
, the above condition is fulfilled by the maximal ideal 
. In fact, the notion
of Cohen-Macaulay ring was inspired by polynomial rings. The above property was proven
for the first time by Macaulay (1916) for every ideal in a polynomial ring over the
complex field.
