Cancellation ideals and minimal cancellation ideals of some commutative rings with identity

Abstract

Let R be a commutative ring with identity. An ideal J of R is called a cancellation ideal if whenever JA=JB for ideals A and B of R, then A=B. And J is called a cancellation ideal belonging to I if J is a cancellation ideal and I J. In this research, we obtain the important results as the two following theorems. 1) Let D be a unique factorization domain and a,b D\{0} such that d=(a,b). Then <a,b> is a cancellation ideal of D if and only if <d>=<a,b>. 2) For all m N, if J is a cancellation ideal belonging to <2,xm> of Z[x], then J=Z[x].