A semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. A bounded semiring is a semiring equipped with a compatible bounded partial order. I...A semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. A bounded semiring is a semiring equipped with a compatible bounded partial order. In this paper, properties of zero divisors and prime elements of a bounded semiring are studied. In particular, it is proved that under some mild assumption, the set Z（A） of nonzero zero divisors of A is A ／ {0, 1}, and each prime element of A is a maximal element. For a bounded semiring A with Z（A） = A ／ {0, 1}, it is proved that A has finitely many maximal elements if ACC holds either for elements of A or for principal annihilating ideals of A. As an application of prime elements, we show that the structure of a bounded semiring A is completely determined by the structure of integral bounded semirings if either ｜Z（A）｜ = 1 or ｜Z（A）｜ -- 2 and Z（A）2 ≠ 0. Applications to the ideal structure of commutative rings are also considered. In particular, when R has a finite number of ideals, it is shown that the chain complex of the poset I（R） is pure and shellable, where I（R） consists of all ideals of R.展开更多
Let G be a finite abelian group and S be a sequence with elements of G.We say that S is a regular sequence over G if|SH|≤|H|-1 holds for every proper subgroup H of G,where SH denotes the subsequence of S consisting o...Let G be a finite abelian group and S be a sequence with elements of G.We say that S is a regular sequence over G if|SH|≤|H|-1 holds for every proper subgroup H of G,where SH denotes the subsequence of S consisting of all terms of S contained in H.We say that S is a zero-sum free sequence over G if 0■Σ(S),where Σ(S)■G denotes the set of group elements which can be expressed as a sum of a nonempty subsequence of S.In this paper,we study the inverse problems associated with Σ(S)when S is a regular sequence or a zero-sum free sequence over G=Cp■Cp,where p is a prime.展开更多
A ring with involution ＊ is called ＊-clean if each of its elements is the sum of a unit and a projection. It is obvious that ＊-clean rings are clean. Vas asked whether there exists a clean ring with involution that...A ring with involution ＊ is called ＊-clean if each of its elements is the sum of a unit and a projection. It is obvious that ＊-clean rings are clean. Vas asked whether there exists a clean ring with involution that is not ＊-clean. In this paper, we investigate when a group ring RG is ＊-clean, where ＊ is the classical involution on RG. We obtain necessary and sufficient conditions for RG to be ＊-clean, where R is a commutative local ring and G is one of the groups C3,C4, S3 and Q8. As a consequence, we provide many examples of group rings which are clean but not ＊-clean.展开更多
基金Acknowledgements This work was supported by the National Natural Science Foundation of China （Grant No. 11271250）.
文摘A semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. A bounded semiring is a semiring equipped with a compatible bounded partial order. In this paper, properties of zero divisors and prime elements of a bounded semiring are studied. In particular, it is proved that under some mild assumption, the set Z（A） of nonzero zero divisors of A is A ／ {0, 1}, and each prime element of A is a maximal element. For a bounded semiring A with Z（A） = A ／ {0, 1}, it is proved that A has finitely many maximal elements if ACC holds either for elements of A or for principal annihilating ideals of A. As an application of prime elements, we show that the structure of a bounded semiring A is completely determined by the structure of integral bounded semirings if either ｜Z（A）｜ = 1 or ｜Z（A）｜ -- 2 and Z（A）2 ≠ 0. Applications to the ideal structure of commutative rings are also considered. In particular, when R has a finite number of ideals, it is shown that the chain complex of the poset I（R） is pure and shellable, where I（R） consists of all ideals of R.
基金supported in part by the Fundamental Research Funds for the Central Universities(No.3122019152)the National Natural Science Foundation of China(Grant Nos.11701256,11871258)+2 种基金the Youth Backbone Teacher Foundation of Henan's University(No.2019GGJS196)the China Scholarship Council(Grant No.201908410132)was also supported in part by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada(Grant No.RGPIN 2017-03903).
文摘Let G be a finite abelian group and S be a sequence with elements of G.We say that S is a regular sequence over G if|SH|≤|H|-1 holds for every proper subgroup H of G,where SH denotes the subsequence of S consisting of all terms of S contained in H.We say that S is a zero-sum free sequence over G if 0■Σ(S),where Σ(S)■G denotes the set of group elements which can be expressed as a sum of a nonempty subsequence of S.In this paper,we study the inverse problems associated with Σ(S)when S is a regular sequence or a zero-sum free sequence over G=Cp■Cp,where p is a prime.
基金This research was supported in part by the National Natural Science Foundation of China （11371089, 11201064）, the Specialized Research Fund for the Doctoral Program of Higher Education （20120092110020）, the Natural Science Foundation of Jiangsu Province （BK20130599）, and a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.
文摘A ring with involution ＊ is called ＊-clean if each of its elements is the sum of a unit and a projection. It is obvious that ＊-clean rings are clean. Vas asked whether there exists a clean ring with involution that is not ＊-clean. In this paper, we investigate when a group ring RG is ＊-clean, where ＊ is the classical involution on RG. We obtain necessary and sufficient conditions for RG to be ＊-clean, where R is a commutative local ring and G is one of the groups C3,C4, S3 and Q8. As a consequence, we provide many examples of group rings which are clean but not ＊-clean.
