(1997). On the spectrum of a module over a commutative ring. Communications in Algebra: Vol. 25, No. 1, pp. 79-103.

On the spectrum of a module over a commutative ring

By considering the notion of multiplication modules over a
commutative ring with identity, first we introduce the notion
product of two submodules of such modules. Then we use this
notion to characterize the prime submodules of a multiplication
module. Finally, we state and prove a version of Nakayama lemma
for multiplication modules and find some related basic results.

/pdf/on-the-prime-submodules-of-multiplication-modules-24evv595ve.pdf

On the prime submodules of multiplication modules

The radical of a module over a commutative ring is the intersection of all prime submodules. It is proved that if R is a commutative domain which is either Noetherian or a UFD then R is one-dimensional if and only if every (finitely generated) primary R-module has prime radical, and this holds precisely when every (finitely generated) R-module satisfies the radical formula for primary submodules.

/pdf/primary-modules-over-commutative-rings-i7jvbuv5p7.pdf

Primary modules over commutative rings

Let M be a left R-module. Then a proper submodule P of M is called weakly prime submodule if for any ideals A and B of R and any submodule N of M such that ABN ⊆ P, we have AN ⊆ P or BN ⊆ P. We define weakly prime radicals of modules and show that for Ore domains, the study of weakly prime radicals of general modules reduces to that of torsion modules. We determine the weakly prime radical of any module over a commutative domain R with dim (R) ≦ 1. Also, we show that over a commutative domain R with dim (R) ≦ 1, every semiprime submodule of any module is an intersection of weakly prime submodules. Localization of a module over a commutative ring preserves the weakly prime property. An R-module M is called semi-compatible if every weakly prime submodule of M is an intersection of prime submodules. Also, a ring R is called semi-compatible if every R-module is semi-compatible. It is shown that any projective module over a commutative ring is semi-compatible and that a commutative Noetherian ring R is semi-compatible if and only if for every prime ideal B of R, the ring R/\B is a Dedekind domain. Finally, we show that if R is a UFD such that the free R-module R⊕ R is a semi-compatible module, then R is a Bezout domain.

On weakly prime radical of modules and semi-compatible modules

Let M be a left R-module. A proper submodule P of M is called classical prime if for all ideals and for all submodules N ⊆ M, implies that or . We generalize the Baer–McCoy radical (or classical prime radical) for a module [denoted by cl.radR(M)] and Baer's lower nilradical for a module [denoted by Nil*(RM)]. For a module RM, cl.radR(M) is defined to be the intersection of all classical prime submodules of M and Nil*(RM) is defined to be the set of all strongly nilpotent elements of M (defined later). It is shown that, for any projective R-module M, cl.radR(M) = Nil*(RM) and, for any module M over a left Artinian ring R, cl.radR(M) = Nil*(RM) = Rad(M) = Jac(R)M. In particular, if R is a commutative Noetherian domain with dim(R) ≤ 1, then for any module M, we have cl.radR(M) = Nil*(RM). We show that over a left bounded prime left Goldie ring, the study of Baer–McCoy radicals of general modules reduces to that of torsion modules. Moreover, over an FBN prime ring R with dim(R) ≤ 1 (or over a commutative domain R with dim(R) ≤ 1), every semiprime submodule of any module is an intersection of classical prime submodules.

A generalization of baer's lower nilradical for modules

(1992). On the prime radical of a module over a commutative ring. Communications in Algebra: Vol. 20, No. 12, pp. 3593-3602.

James Jenkins

Papers

On the prime radical of a module over a commutative ring