Divisibility Properties of the Semiring of Ideals of an Integral Domain

• 作者: Gyu Whan Chang HwanKoo Kim
• 来源期刊: 代数集刊（英文版）
• 年，卷（期）: 2020,（3）
• 页码范围: 369-380
• 页数: 12页
• 语种: 英文
• DOI: 10.1142/S1005386720000309

摘要

Let D be an integral domain,F+ (D) (resp.,f+ (D)) be the set of nonzero (resp.,nonzero finitely generated) ideals of D,R1 =f+(D) ∪ {(0)},and R2 =F+(D) ∪ {(0)}.Then (Ri,⊕,(X)) for i =1,2 is a commutative semiring with identity under I ⊕ J =I + J and I (X) J =IJ for all I,J ∈Ri.In this paper,among other things,we show that D is a Prüfer domain if and only if every ideal of R1 is a k-ideal if and only if R1 is Gaussian.We also show that D is a Dedekind domain if and only if R2 is a unique factorization semidomain if and only if R2 is a principal ideal semidomain.These results are proved in a more general setting of star operations on D.