Strong Irreducible Ideal Completely

In this paper, the concepts of completely irreducible ideals and completely strongly irreducible ideals in a commutative ring are introduced, which are regarded as special cases of the notions of W.Grobner’s irreducible ideals and L.Fuchs’s strongly irreducible ideals, respectively. We investigate the relations between completely irreducible ideals and prime ideals, and prove that if R is a ring, x∈R, then x is not nilpotent, if and only if there exists a value M (completely irreducible ideals) of x, such that M is a prime ideal. Secondly, a equivalent describe of completely strongly irreducible ideals is proposed, that is, an ideal M is completely strongly irreducible ideals, if and only if M is unique value of a non zero element of R. In particular, if J(R)=0, then M is completely strongly irreducible ideals, if and only if there exists0≠e∈Idem(R), such that M is unique value of e. Thirdly, based on the relations between completely irreducible ideals and completely strongly irreducible ideals, a class of special rings, which is referred to as completely arithmetical rings, are considered as following:a ring R is said to be a completely arithmetical ring, if for any ideal I of R and any nonempty family of ideals {Kλ|λ∈(?)} of R, we have ((?)λ∈(?)Kλ\+I=λ∈(?)(Kλ+I). By an example we shall show that the class of completely arithmetical rings is, in fact, a proper subclass of arithmetical rings. Finally, according to the properties of completely irreducible ideals and completely strongly irreducible ideals, another necessary and sufficient condition is presented, that is, a ring R is a completely arithmetical ring if and only if every completely irreducible ideal of R is a completely strongly irreducible ideal.In particular, we obtain that a ring R is a completely arithmetical ring, and J(R)=0, if and only if R is regular and semi local ring.