2012/02/02 by Mahmood Behboodi, Behboodi, Mahmood, Masoud Sabzevari +1
Mathematics · #13A99 #13C13 #13C99 #14A25 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #math.AC #math.AG #math.RA #msc:13A99 #msc:13C13 #msc:13C99 #msc:14A25
paper · pdf · doi:10.48550/arxiv.1202.0377
18 Pages
arxiv created 2012/02/02 · openalex publication_date 2012/02/02 · arxiv updated 2012/02/03 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
The purpose of this paper and its sequel, is to introduce a new class of modules over a commutative ring R, called ℙ-radical modules (modules M satisfying the prime radical condition "(√[p]\calPM:M)=\calP" for every prime ideal \calP⊇ \rm Ann(M), where √[p]\calPM is the intersection of all prime submodules of M containing \calPM). 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 ℙ-radical modules. Also, we show that if R is a domain (or a Noetherian ring), then all projective modules are ℙ-radical. In particular, if R is an Artinian ring, then all R-modules are ℙ-radical and the converse is also true when R is a Noetherian ring. Also an R-module M is called \mathbbM-radical if (√[p]\calMM:M)=\calM; for every maximal ideal \calM⊇ \rm Ann(M). We show that the two concepts ℙ-radical and \mathbbM-radical are equivalent for all R-modules if and only if R is a Hilbert ring. Semisimple ℙ-radical (\mathbbM-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 ℙ-radical modules (with the Zariski topology) resembles to that of rings.