Let R be a commutative ring (with 1) and let Z(R) be its set of zero-divisors. The zero-divisor graph Γ(R) has vertex set Z∗(R) = Z(R) \ {0} and for distinct x, y ∈ Z∗(R), the vertices x and y are adjacent if and only if xy = 0. In this paper, we consider the domination number and signed domination number on zero-divisor graph Γ(R) of commutative ring R such that for every 0 6= x ∈ Z∗(R), x 6= ...

We address two aspects of finitely generated modules of finite projective dimension over local rings and their connection in between: embeddability and grade of order ideals of minimal generators of syzygies. We provide a solution of the embeddability problem and prove important reductions and special cases of the order ideal conjecture. In particular we derive that in any local ring R of mixed...

We determine the automorphism group of zero-divisor digraph semiring matrices over an antinegative commutative with a finite number zero-divisors.

For a smooth scheme X of pure dimension d over field k and an effective Cartier divisor D⊂X whose support is simple normal crossing divisor, we construct cycle class map from the Chow group zero-cycles with modulus to cohomology relative Milnor K-sheaf.

For a smooth scheme X of pure dimension d over field k and an effective Cartier divisor D⊂X whose support is simple normal crossing divisor, we construct cycle class map from the Chow group zero-cycles with modulus to cohomology relative Milnor K-sheaf.

Elementary divisor domains were defined by Kaplansky [I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc. 66 (1949) 464–491] and generalized to rings with zero-divisors by Gillman and Henriksen [L. Gillman, M. Henriksen, Some remarks about elementary divisor rings, Trans. Amer. Math. Soc. 82 (1956) 362–365]. In [M.D. Larsen, W.J. Lewis, T.S. Shores, Elementary divisor rings a...