A classification of primary ideals in a non commutative valuation ring
by
Hidetoshi Marubayashi
Dept. of Mathematics, Naruto University of Education, Naruto, 772, Japan

An ideal A of a non comutative valuation ring R with P = \surdA is called right P-primary if aRb subset or equal A (a, b in R implies either a in A or b in P, where \surdA is the prime radical of A.

A classification of right P-primary ideals deeply depend on the prime segments and the right order of A as follows:

(P1) there is a Goldie prime ideals P₁ such that P subset or equal P₁ and it is a segment.

(a) The segment P subset or equal P₁ is Archimedean.

(b) The segment P subset or equal C subset or equal P₁ is exceptional, where C is the non Goldie prime ideal.

(c) The segment p subset or equal P₁ is simple.

(P2) P is a lower limit prime ideal.

The main results are:

Theorem 1. (1) I case of one of (P1) (a), (P1)(b) and (P2), A is right P-primary if and only if O_r(A) = { q in Q | Aq subset or equal A }, the right order of A , is equal to R, where Q is the quotient ring of R which is simple Artinian.

(2) In case of (P1)(c), A is right P-primary iff either O_r(A) = R_P or O_r(A) = R_P1 and A = A^*, where A^* = A^-1-1.

(3) { (Cⁿ)^* | n = 1, 2, ... , } is the set of all right C-primary ideals.

The types of primary ideals A satisfying O_r (A) = R_P1 and A = A^* and of (Cⁿ)^* are typical ones in nonnnn commutative vcaluation rings which never appear in commutative valuation rings, even P. I. non commutative valuation rings. It is showen by using Theorem 1 that a right p-primary ideals A is one of the following:

Theorem 2. A = aR_P for some a in A, A = aP for some a in A^*, A = \cap cR_P (A subset or equal cR_P, c in Q), A = aR_P1 for some a in A and A = \cap cR_P1 ( A subset or equal cR_P1, c in Q), where the segment P subset or equal P₁ is simple.

Examples of right P-primary ideal A satisfying O_r(A) + R_P1 and A = A^* are given and seversal open problems are also discussed.

Date received: May 3, 1999