EINE CHARAKTERISIERUNG DER MENGE Ass$(\frak U)$ IN KOMMUTATIVEN NOETHERSCHEN RINGEN

Veselin Peri\'c

Abstract: Let $R$ be a commutative noetherian ring. For any ideal $\frak U$ of $R$ the set Ass$(\frak U)$ of all associated prime ideals of $\frak U$ is a finite set $P$ of prime ideals of $R$. If a finite set $P$ of prime ideals of $R$ contains no prime ideal of the height 0, i.e., no minimal prime ideal of $(0)$, then it is well known that there exists an ideal $\frak U$ of $R$ such that $P= \text{Ass}(\frak U)$ [1, 9.1]. It seems to be unknown what the precise necessary and sufficient condition is, on a finite set $P$ of primes, for the existence of such an ideal $\frak U$ [1, p. 68]. We answer here this question.

Keywords: Noetherian commutative rings, primary decomposition of ideals, associated prime ideals

Classification (MSC2000): 13A17, 13E05

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