In this paper,we consider Ramanujan’s sums over arbitrary Dedekind domain with finite norm property.We define the Ramanujan’s sumsη(a,A)andη(B,A),where a is an arbitrary element in a Dedekind domain,B is an ideal ...In this paper,we consider Ramanujan’s sums over arbitrary Dedekind domain with finite norm property.We define the Ramanujan’s sumsη(a,A)andη(B,A),where a is an arbitrary element in a Dedekind domain,B is an ideal and A is a non-zero ideal.In particular,we discuss the Kluyver formula and Hèolder formula forη(a,A)andη(B,A).We also prove the reciprocity formula enjoyed byη(B,A)and the orthogonality relations forη(a,A)in the last two parts.展开更多
We give an algorithm for computing the factor ring of a given ideal in a Dedekind domain with finite rank, which runs in deterministic and polynomial time. We provide two applications of the algorithm:judging whether ...We give an algorithm for computing the factor ring of a given ideal in a Dedekind domain with finite rank, which runs in deterministic and polynomial time. We provide two applications of the algorithm:judging whether a given ideal is prime or prime power. The main algorithm is based on basis representation of finite rings which is computed via Hermite and Smith normal forms.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 R be a domain.In this paper,we show that if R is one dimensional,then R is a Noetherian Warfield domain if and only if every maximal ideal of R is 2-generated and for every maximal ideal M of R,M is divisorial in ...Let R be a domain.In this paper,we show that if R is one dimensional,then R is a Noetherian Warfield domain if and only if every maximal ideal of R is 2-generated and for every maximal ideal M of R,M is divisorial in the ring(M:M).We also prove that a Noetherian domain R is a Noetherian Warfield domain if and only if for every maximal ideal M of R,M^(2) can be generated by two elements.Finally,we give a sufficient condition under which all ideals of R are strongly Gorenstein projective.展开更多
基金Supported by the National Research and Development Program of China(Grant No.2018YFB1107402)。
文摘In this paper,we consider Ramanujan’s sums over arbitrary Dedekind domain with finite norm property.We define the Ramanujan’s sumsη(a,A)andη(B,A),where a is an arbitrary element in a Dedekind domain,B is an ideal and A is a non-zero ideal.In particular,we discuss the Kluyver formula and Hèolder formula forη(a,A)andη(B,A).We also prove the reciprocity formula enjoyed byη(B,A)and the orthogonality relations forη(a,A)in the last two parts.
基金supported by National Natural Science Foundation of China (Grant Nos. 11601202, 11471314 and 11401312)the Natural Science Foundation of the Jiangsu Higher Education Institutions (Grant No. 14KJB110012)+1 种基金the High-Level Talent Scientific Research Foundation of Jinling Institute of Technology (Grant No. jit-b-201527)the National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences
文摘We give an algorithm for computing the factor ring of a given ideal in a Dedekind domain with finite rank, which runs in deterministic and polynomial time. We provide two applications of the algorithm:judging whether a given ideal is prime or prime power. The main algorithm is based on basis representation of finite rings which is computed via Hermite and Smith normal forms.
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.
基金This work was partially supported by the Department of Mathematics in Kyungpook National University and National Natural Science Foundation of China(Grant No.11671283)The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea(NRF)funded by the Ministry of Education,Science and Technology(2017R1C1B1008085),Korea.
文摘Let R be a domain.In this paper,we show that if R is one dimensional,then R is a Noetherian Warfield domain if and only if every maximal ideal of R is 2-generated and for every maximal ideal M of R,M is divisorial in the ring(M:M).We also prove that a Noetherian domain R is a Noetherian Warfield domain if and only if for every maximal ideal M of R,M^(2) can be generated by two elements.Finally,we give a sufficient condition under which all ideals of R are strongly Gorenstein projective.
