A graph property is any class of graphs that is closed under isomorphisms, A graph property P is hereditary if it is closed under taking subgraphs; it is compositive if for any graphs
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
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
The concept of an I-matrix in the full 2 × 2 matrix ring M2 （R/I）, where R is an arbitrary UFD and I is a nonzero ideal in R, is introduced. We obtain a concrete description of the centralizer of an/-matrix B i...The concept of an I-matrix in the full 2 × 2 matrix ring M2 （R/I）, where R is an arbitrary UFD and I is a nonzero ideal in R, is introduced. We obtain a concrete description of the centralizer of an/-matrix B in M_2（R/I） as the sum of two subrings ,S_1 and ,S_2 of M_2（R/I）, where S_1 is the image （under the natural epimorphism from M_2（R） to M_2（R/I）） of the centralizer in M_2（R） of a pre-image of B, and the entries in S_2 are intersections of certain annihilators of elements arising from the entries of B. It turns out that if R is a PID, then every matrix in M_2（R/I） is an/-matrix. However, this is not the case if R is a UFD in general. Moreover, for every factor ring R/I with zero divisors and every n ≥ 3, there is a matrix for which the mentioned concrete description is not valid.展开更多
文摘A graph property is any class of graphs that is closed under isomorphisms, A graph property P is hereditary if it is closed under taking subgraphs; it is compositive if for any graphs
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.
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.
文摘The concept of an I-matrix in the full 2 × 2 matrix ring M2 （R/I）, where R is an arbitrary UFD and I is a nonzero ideal in R, is introduced. We obtain a concrete description of the centralizer of an/-matrix B in M_2（R/I） as the sum of two subrings ,S_1 and ,S_2 of M_2（R/I）, where S_1 is the image （under the natural epimorphism from M_2（R） to M_2（R/I）） of the centralizer in M_2（R） of a pre-image of B, and the entries in S_2 are intersections of certain annihilators of elements arising from the entries of B. It turns out that if R is a PID, then every matrix in M_2（R/I） is an/-matrix. However, this is not the case if R is a UFD in general. Moreover, for every factor ring R/I with zero divisors and every n ≥ 3, there is a matrix for which the mentioned concrete description is not valid.
