ON COMMUTATIVE RINGS WHOSE PRIME IDEALS ARE DIRECT SUMS OF CYCLICS

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, M), the following statements are equivalent: (1) Every prime ideal of R is a direct sum of cyclic R-modules; (2) M = circle plus(lambda is an element of Lambda) R omega(lambda) where Lambda is an index set and R/Ann(w(lambda)) is a principal ideal ring for each lambda is an element of Lambda; (3) Every prime ideal of R is a direct sum of at most vertical bar Lambda vertical bar cyclic R-modules where Lambda is an index set and M = circle plus(lambda is an element of Lambda) R-omega lambda ; 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, M) is a direct sum of (at most n) principal ideals, it suffices to test only the maximal ideal M.