Reduced Noetherian one-dimensional implies Cohen-Macaulay

Statement

Verbal statement

A reduced Noetherian one-dimensional ring is Cohen-Macaulay.

Definitions used

Noetherian ring

Further information: Noetherian ring

A commutative unital ring is termed Noetherian if every ideal in it is finitely generated.

Reduced ring

Further information: reduced ring

A commutative unital ring is termed reduced if it has no nonzero nilpotent element, i.e. its nilradical is zero. In other words, the intersection of all prime ideals is zero.

For a Noetherian ring, this is equivalent to demanding that the intersection of all the minimal primes equals zero.