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
"(pPMβ:M)=P" for every prime ideal
PβAnn(M), where pPMβ is the
intersection of all prime submodules of M containing PM). 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 M-radical if
(pMMβ:M)=M; for every maximal ideal
MβAnn(M). We show that the two concepts
P-radical and M-radical are equivalent for all
R-modules if and only if R is a Hilbert ring. Semisimple
P-radical (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 P-radical
modules (with the Zariski topology) resembles to that of rings.Comment: 18 Page