On Commutative Rings Whose Prime Ideals Are Direct Sums of Cyclics

In this paper we study commutative rings $R$ whose prime ideals are direct sums of cyclic modules. In the case $R$ is a finite direct product of commutative local rings, the structure of such rings is completely described. In particular, it is shown that for a local ring $(R, \cal{M})$, the following statements are equivalent: (1) Every prime ideal of $R$ is a direct sum of cyclic $R$-modules; (2) ${\cal{M}}=\bigoplus_{\lambda\in \Lambda}Rw_{\lambda}$ and $R/{\rm Ann}(w_{\lambda})$ is a principal ideal ring for each $\lambda \in \Lambda$;(3) Every prime ideal of $R$ is a direct sum of at most $|\Lambda|$ cyclic $R$-modules; and (4) Every prime ideal of $R$ is a summand of a direct sum of cyclic $R$-modules. Also, we establish a theorem which state that, to check whether every prime ideal in a Noetherian local ring $(R, \cal{M})$ is a direct sum of (at most $n$) principal ideals, it suffices to test only the maximal ideal $\cal{M}$.Comment: 9 Page

A prime ideal principle for two-sided ideals

Many classical ring-theoretic results state that an ideal that is maximal with respect to satisfying a special property must be prime. We present a "Prime Ideal Principle" that gives a uniform method of proving such facts, generalizing the Prime Ideal Principle for commutative rings due to T.Y. Lam and the author. Old and new "maximal implies prime" results are presented, with results touching on annihilator ideals, polynomial identity rings, the Artin-Rees property, Dedekind-finite rings, principal ideals generated by normal elements, strongly noetherian algebras, and just infinite algebras.Comment: 22 page

Examples of non-Noetherian domains inside power series rings

Let R* be an ideal-adic completion of a Noetherian integral domain R and let L be a subfield of the total quotient ring of R* such that L contains R. Let A denote the intersection of L with R*. The integral domain A sometimes inherits nice properties from R* such as the Noetherian property. For certain fields L it is possible to approximate A using a localzation B of a nested union of polynomial rings over R associated to A; if B is Noetherian, then B = A. If B is not Noetherian, we can sometimes identify the prime ideals of B that are not finitely generated. We have obtained in this way, for each positive integer s, a 3-dimensional local unique factorization domain B such that the maximal ideal of B is 2-generated, B has precisely s prime ideals of height 2, each prime ideal of B of height 2 is not finitely generated and all the other prime ideals of B are finitely generated. We examine the map Spec A to Spec B for this example. We also present a generalization of this example to dimension 4. We describe a 4-dimensional local non-Noetherian UFD B such that the maximal ideal of B is 3-generated, there exists precisely one prime ideal Q of B of height 3, the prime ideal Q is not finitely generated. We consider the question of whether Q is the only prime ideal of B that is not finitely generated, but have not answered this question.Comment: 32 pages to appear in JC