Divisibility Proper ties of the Semiring of Ideals of an Integral Domain  

作　　者：Gyu Whan Chang HwanKoo Kim 

机构地区：[1]Department of Mathematics Education,Incheon National University Incheon 22012,Korea [2]Division of Computer and Information Engineering Hoseo University,Asan 336-795,Korea

出　　处：《Algebra Colloquium》2020年第3期369-380,共12页代数集刊（英文版）

基　　金：supported by the Incheon National University Research Grant in 2018(Grant No.2018-0014).

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,R_(1)=f+(D)∪{(0)},and R_(2)=F+(D)∪{(0)}.Then(R_(i),㊉,■)for i=1,2 is a commutative semiring with identity under I㊉J=I+J and I■J=ZJ for all I,J∈R_(i).In this paper,among other things,we show that D is a Priifer domain if and only if every ideal of R_(1)is a k-ideal if and only if R_(1)is Gaussian.We also show that D is a Dedekind domain if and only if R_(2)is a unique factorization semidomain if and only if R_(2)is a principal ideal semidomain.These results are proved in a more general setting of star operations on D.

关 键 词：star operation P*MD *-Dedekind domain Gaussian semiring Bezout semiring unique factorization semidomain principal ideal semidomain 

分 类 号：O15[理学—数学]