26 research outputs found
On Commutative Rings Whose Prime Ideals Are Direct Sums of Cyclics
In this paper we study commutative rings whose prime ideals are direct
sums of cyclic modules. In the case 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 , the following
statements are equivalent: (1) Every prime ideal of is a direct sum of
cyclic -modules; (2)
and is a principal ideal ring for each ;(3) Every prime ideal of is a direct sum of at most
cyclic -modules; and (4) Every prime ideal of is a summand of a direct
sum of cyclic -modules. Also, we establish a theorem which state that, to
check whether every prime ideal in a Noetherian local ring is a
direct sum of (at most ) principal ideals, it suffices to test only the
maximal ideal .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 , called -radical modules
(modules satisfying the prime radical condition
"" for every prime ideal
, where is the
intersection of all prime submodules of containing ). 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 -radical modules. Also, we show that if is a domain (or a
Noetherian ring), then all projective modules are -radical. In
particular, if is an Artinian ring, then all -modules are
-radical and the converse is also true when is a Noetherian
ring. Also an -module is called -radical if
; for every maximal ideal
. We show that the two concepts
-radical and -radical are equivalent for all
-modules if and only if is a Hilbert ring. Semisimple
-radical (-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 -radical
modules (with the Zariski topology) resembles to that of rings.Comment: 18 Page