Distribution Ring Characterized

In this paper, we first establish a characterization of arithemetical rings, i.e., we prove that a ring R is arithmetical if and only if every irreducible ideal of R is strongly irreducible. Then we gave the definition of transitive ideal in a ring and establish extension of arithemetical rings, i.e., we prove that a ring R is arithmetical if and only if there exists s a transitive ideal I of R such that I and R/I are both arithmetical. Next we proved some properties of arithmetical rings. R is an arithmetical ring if and only if R [x] is an arithmetical ring. If R is an arithmetical ring,Sis a multiplicatively closed set in R, then S-1 Ris an arithmetical ring. LetMn(R)be the ring of upper triangularn×n matrices with entries in R, ifMn(R)is an arithmetical ring, then R is an arithmetical ring. If R is an arithmetical ring, then R is an Abelian ring. At last, we discussed the relationships among the family of prime ideal, strongly prime ideal, strongly irreducible ring and irreducible ideal for any ring.