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

Modules Satisfying the Prime Radical Condition and a Sheaf Construction for Modules I

The purpose of this paper and its sequel, is to introduce a new class of modules over a commutative ring $R$, called $\mathbb{P}$-radical modules (modules $M$ satisfying the prime radical condition "$(\sqrt[p]{{\cal{P}}M}:M)={\cal{P}}$" for every prime ideal ${\cal{P}}\supseteq {\rm Ann}(M)$, where $\sqrt[p]{{\cal{P}}M}$ is the intersection of all prime submodules of $M$ containing ${\cal{P}}M$). This class contains the family of primeful modules properly. This yields that over any ring all free modules and all finitely generated modules lie in the class of $\mathbb{P}$-radical modules. Also, we show that if $R$ is a domain (or a Noetherian ring), then all projective modules are $\mathbb{P}$-radical. In particular, if $R$ is an Artinian ring, then all $R$-modules are $\mathbb{P}$-radical and the converse is also true when $R$ is a Noetherian ring. Also an $R$-module $M$ is called $\mathbb{M}$-radical if $(\sqrt[p]{{\cal{M}}M}:M)={\cal{M}}$; for every maximal ideal ${\cal{M}}\supseteq {\rm Ann}(M)$. We show that the two concepts $\mathbb{P}$-radical and $\mathbb{M}$-radical are equivalent for all $R$-modules if and only if $R$ is a Hilbert ring. Semisimple $\mathbb{P}$-radical ($\mathbb{M}$-radical) modules are also characterized. In Part II we shall continue the study of this construction, and as an application, we show that the sheaf theory of spectrum of $\mathbb{P}$-radical modules (with the Zariski topology) resembles to that of rings.Comment: 18 Page