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
The famous strongly binary Goldbach’s conjecture asserts that every even number 2n ≥ 8 can always be expressible as a sum of two distinct odd prime numbers. We use a new approach to dealing with this conjecture. Spe...The famous strongly binary Goldbach’s conjecture asserts that every even number 2n ≥ 8 can always be expressible as a sum of two distinct odd prime numbers. We use a new approach to dealing with this conjecture. Specifically, we apply the element order prime graphs of alternating groups of degrees 2n and 2n −1 to characterize this conjecture, and present its six group-theoretic versions;and further prove that this conjecture is true for p +1 and p −1 whenever p ≥ 11 is a prime number.展开更多
In this paper, we give the conception of implicit congruence and nonimplicit congruence in a unique factorization domain R and establish some structures of irreducible polynomials over R . A classical result, E...In this paper, we give the conception of implicit congruence and nonimplicit congruence in a unique factorization domain R and establish some structures of irreducible polynomials over R . A classical result, Eisenstein′s criterion, is generalized.展开更多
Abstract In the present paper, some basic properties of MP filters of Ro algebra M are investigated. It is proved that（FMP（M）,包含,′∧^-∨^-,{1},M）is a bounded distributive lattice by introducing the negation ope...Abstract In the present paper, some basic properties of MP filters of Ro algebra M are investigated. It is proved that（FMP（M）,包含,′∧^-∨^-,{1},M）is a bounded distributive lattice by introducing the negation operator ′, the meet operator ∧^-, the join operator ∨^- and the implicati on operator → on the set FMP（M） of all MP filters of M. Moreover, some conditions under which （FMP（M）,包含,′∨^-,→{1},M）is an Ro algebra are given. And the relationship between prime elements of FMP （M） and prime filters of M is studied. Finally, some equivalent characterizations of prime elements of .FMP （M） are obtained.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.
文摘The famous strongly binary Goldbach’s conjecture asserts that every even number 2n ≥ 8 can always be expressible as a sum of two distinct odd prime numbers. We use a new approach to dealing with this conjecture. Specifically, we apply the element order prime graphs of alternating groups of degrees 2n and 2n −1 to characterize this conjecture, and present its six group-theoretic versions;and further prove that this conjecture is true for p +1 and p −1 whenever p ≥ 11 is a prime number.
文摘In this paper, we give the conception of implicit congruence and nonimplicit congruence in a unique factorization domain R and establish some structures of irreducible polynomials over R . A classical result, Eisenstein′s criterion, is generalized.
文摘Abstract In the present paper, some basic properties of MP filters of Ro algebra M are investigated. It is proved that（FMP（M）,包含,′∧^-∨^-,{1},M）is a bounded distributive lattice by introducing the negation operator ′, the meet operator ∧^-, the join operator ∨^- and the implicati on operator → on the set FMP（M） of all MP filters of M. Moreover, some conditions under which （FMP（M）,包含,′∨^-,→{1},M）is an Ro algebra are given. And the relationship between prime elements of FMP （M） and prime filters of M is studied. Finally, some equivalent characterizations of prime elements of .FMP （M） are obtained.
