One-dimensional ring

Definition
A commutative unital ring is termed one-dimensional if it satisfies the following equivalent conditions:


 * Its Krull dimension is one
 * Every prime ideal in it is either a minimal prime ideal or a maximal ideal, and not every prime ideal is both (i.e. there exists at least one minimal prime ideal that is not maximal, or at least one maximal ideal that is not a minimal prime)

Conjunction with other properties

 * One-dimensional domain: This is a one-dimensional ring where there is a unique minimal prime ideal: the zero ideal
 * One-dimensional local ring: This is a one-dimensional ring with a unique maximal ideal
 * One-dimensional Noetherian ring: A one-dimensional ring that is also Noetherian
 * One-dimensional reduced ring: This is a one-dimensional ring where the intersection of all minimal primes is zero