Cohen-Macaulay module
A module over a ring is a Cohen-Macaulay module if its depth is defined and equals its Krull dimension. A ring is said to be Cohen-Macaulay (or just C-M) if it is a Cohen-Macaulay module viewed as a module over itself.
Cohen-Macaulay rings are used extensively in combinatorial geometry and commutative ring theory, and has applications to algebraic geometry as well. For instance, a variety all of whose local rings
are Cohen-Macaulay has, in a sense, nicer behaviour than an arbitrary singular variety.