Irreducibility And Cancellation Of Ideals

Completely strongly irreducible ideal is a natural extension of strongly irreducible ideal from finite to infinite in[1].It is also the enhancement of completely irreducible ideal[2].In this paper,we establish a characterization about completely strongly ir-reducible and investigate the relationship between completely irreducible ideals and prime ideals.We also investigate the relationship between completely strongly irre-ducible ideal and completely irreducible ideal.We get the conclusion that if R is a ring with J(R)= 0,then each completely irreducible ideal is completely strongly irreducible if and only if R is both regular and semiperfect.This relationship leads to the creation of a new ring-completely arithmetical ring.Furthermore,we investigate the structure of the ring on the basis of regular ring.In the second part of the paper,we mainly investigate cancellation of ideal.It is a very active topic in commutative algebra.We establish a series of characterizations in regular arithmetical ring which provides a basis for further research on cancellation of ideal.We can apply this result to learn cancellation of ideal clearly.And then we can get an equivalent condition:If R is a completely arithmetical ring with J(R)= 0,then N is a cancellation ideal if and only if for any e? Idem(R),there exists f?Idem(N) such that Re = Rf.