2012/02/02 by Mahmood Behboodi, Behboodi, Mahmood, Ali Moradzadeh-Dehkordi +1
Mathematics · #13E05 #13F10 #13H99 #Commutative Algebra (math.AC) #FOS: Mathematics #Primary 13C05 #Rings and Algebras (math.RA) #Secondary 13E10 #math.AC #math.RA #msc:13C05 #msc:13E05 #msc:13E10 #msc:13F10 #msc:13H99
paper · pdf · doi:10.48550/arxiv.1202.0371
9 Pages
arxiv created 2012/02/02 · arxiv updated 2012/02/03
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, \calM), the following statements are equivalent: (1) Every prime ideal of R is a direct sum of cyclic R-modules; (2) \calM=\bigoplusλ∈ ΛRwλ and R/\rm Ann(wλ) is a principal ideal ring for each λ∈ Λ;(3) Every prime ideal of R is a direct sum of at most |Λ| 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, \calM) is a direct sum of (at most n) principal ideals, it suffices to test only the maximal ideal \calM.