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 wit...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.展开更多
Let D be an integral domain with quotient field K,D be the integral closure of D in K,and D^[w] be the ω-integral closure of D in K;so D ■ D^[w],and equality holds when D is Noetherian or dim(D)=1.The Mori-Nagata th...Let D be an integral domain with quotient field K,D be the integral closure of D in K,and D^[w] be the ω-integral closure of D in K;so D ■ D^[w],and equality holds when D is Noetherian or dim(D)=1.The Mori-Nagata theorem states that if D is Noetherian,then D is a Krull domain;it has also been investigated when D is a Dedekind domain.We study integral domains D such that D^[w] is a Krull domain.We also provide an example of an integral domain D such that D ■ D ■ D^[w],t-dim(D)=1,D is a Priifer multiplication domain with v-dim(D)=2,and D^[w] is a UFD.展开更多
Let D be an integral domain, ψ（D） （resp., t-ψ（D）） be the set of all valuation （resp., t-valuation） ideals of D, and w-P（D） be the set of primary w-ideals of D. Let D[X] be the polynomial ring over 19, c（f...Let D be an integral domain, ψ（D） （resp., t-ψ（D）） be the set of all valuation （resp., t-valuation） ideals of D, and w-P（D） be the set of primary w-ideals of D. Let D[X] be the polynomial ring over 19, c（f） be the ideal of D generated by the coefficients of f ∈ D[X], and Nv = {f∈ D[X] ｜ c（f）v = D}. In this paper, we study integral domains D in which w-P（D） t-ψ（D）, t-ψ（D） w-P（D）, or t-ψ（D） = w-P（D）. We also study the relationship between t-ψ（D） and ψ（D[X]Nv）, and characterize when t-ψ（A ＋ XB[X]） w-P（A ＋ XB[X]） holds for a proper extension A c B of integral domains.展开更多
Let D be a finite-dimensional integral domain, Spec（D） be the set of prime ideals of D, and SpSS（D） be the set of spectral semistar operations on D. Mimouni gave a complete description for the prime ideal structur...Let D be a finite-dimensional integral domain, Spec（D） be the set of prime ideals of D, and SpSS（D） be the set of spectral semistar operations on D. Mimouni gave a complete description for the prime ideal structure of D with ｜SpSS（D）｜ = n ＋ dim（D） for 1 ≤ n ≤5 except for the quasi-local cases of n = 4, 5. In this paper, we show that there is an integral domain D such that ｜SpSS（D） ｜ = n＋dim（D） for all positive integers n with n ≠ 2. As corollaries, we completely characterize the quasi-local domains D with ｜SpSS（D）｜= n＋dim（D） for n = 4, 5. Furthermore, we also present the lower and upper bounds of ISpSS（D）I when Spee（D） is a finite tree.展开更多
基金supported by the Incheon National University Research Grant in 2018(Grant No.2018-0014).
文摘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.
基金supported by the Academic Research Fund of Hoseo University in 2017(no.2017-0047).
文摘Let D be an integral domain with quotient field K,D be the integral closure of D in K,and D^[w] be the ω-integral closure of D in K;so D ■ D^[w],and equality holds when D is Noetherian or dim(D)=1.The Mori-Nagata theorem states that if D is Noetherian,then D is a Krull domain;it has also been investigated when D is a Dedekind domain.We study integral domains D such that D^[w] is a Krull domain.We also provide an example of an integral domain D such that D ■ D ■ D^[w],t-dim(D)=1,D is a Priifer multiplication domain with v-dim(D)=2,and D^[w] is a UFD.
文摘Let D be an integral domain, ψ（D） （resp., t-ψ（D）） be the set of all valuation （resp., t-valuation） ideals of D, and w-P（D） be the set of primary w-ideals of D. Let D[X] be the polynomial ring over 19, c（f） be the ideal of D generated by the coefficients of f ∈ D[X], and Nv = {f∈ D[X] ｜ c（f）v = D}. In this paper, we study integral domains D in which w-P（D） t-ψ（D）, t-ψ（D） w-P（D）, or t-ψ（D） = w-P（D）. We also study the relationship between t-ψ（D） and ψ（D[X]Nv）, and characterize when t-ψ（A ＋ XB[X]） w-P（A ＋ XB[X]） holds for a proper extension A c B of integral domains.
文摘Let D be a finite-dimensional integral domain, Spec（D） be the set of prime ideals of D, and SpSS（D） be the set of spectral semistar operations on D. Mimouni gave a complete description for the prime ideal structure of D with ｜SpSS（D）｜ = n ＋ dim（D） for 1 ≤ n ≤5 except for the quasi-local cases of n = 4, 5. In this paper, we show that there is an integral domain D such that ｜SpSS（D） ｜ = n＋dim（D） for all positive integers n with n ≠ 2. As corollaries, we completely characterize the quasi-local domains D with ｜SpSS（D）｜= n＋dim（D） for n = 4, 5. Furthermore, we also present the lower and upper bounds of ISpSS（D）I when Spee（D） is a finite tree.
