Abnormalities in Noetherian rings

Abstract

If P ⊆ Q P \subseteq Q are prime ideals in some ring R and if rank Q = rank ( Q / P ) + rank P + k Q = {\text {rank}}(Q/P) + {\text {rank}}\;P + k , then P ⊂ Q P \subset Q is said to be k-abnormal and k is called the degree of abnormality. The paper consists of two examples. The first example is a Noetherian integral domain in which the set of degrees of abnormality is unbounded. Let P be a prime ideal of R and set W = { Q / Q W = \{ Q/Q is a prime ideal and P ⊂ Q P \subset Q is abnormal}. The second example is a local domain such that { k | P ⊂ Q \{ k|P \subset Q is k-abnormal for some Q ∈ W } ≠ { k | P ⊂ Q Q \in W\} \ne \{ k|P \subset Q is k-abnormal for some Q minimal in W}.