Cohen-Macaulay Ring -- from Wolfram MathWorld

Cohen-Macaulay Ring

Also called Macaulay ring, a Cohen Macaulay ring a is a Noetherian commutative unit ring 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.

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).

This entry contributed by Margherita Barile

REFERENCES:

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.

CITE THIS AS:

Barile, Margherita. "Cohen-Macaulay Ring." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/Cohen-MacaulayRing.html