25 - M. Behboodi (with A. Moadzadeh-Dehkordi), On commutative rings whose prime ideals are direct sum of cyclics, Arch. Math. 48 (2012), 291–299.
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 = Sum_λ∈Λ Rw_λ and R/Ann(w_λ) 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,M) is a direct sum of (at most n) principal ideals, it suﬃces to test only the maximal ideal M.