Around Zero-Divisor Graph of Skew Polynomial Rings over Real Matrix 2 by 2

In this paper, we continue to study zero-divisor properties of skew polynomial rings [Formula: see text], where [Formula: see text] is an associative ring equipped with an endomorphism [Formula: see text] and an [Formula: see text]-derivation [Formula: see text]. For an associative ring [Formula: see text], the undirected zero-divisor graph of [Formula: see text] is the graph [Formula: see text] such that the vertices of [Formula: see text] are all the nonzero zero-divisors of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are connected by an edge if and only if [Formula: see text] or [Formula: see text]. As an application of reversible rings, we investigate the interplay between the ring-theoretical properties of a skew polynomial ring [Formula: see text] and the graph-theoretical properties of its zero-divisor graph [Formula: see text]. Our goal in this paper is to give a characterization of the possible diameters of [Formula: see text] in terms of the diameter of [Formula: see text], when the base ring [Formula: see text] is reversible and also have the [Formula: see text]-compatible property. We also completely describe the associative rings all of whose zero-divisor graphs of skew polynomials are complete.

Download Full-text

PI degree parity in q-skew polynomial rings

Journal of Algebra ◽

10.1016/j.jalgebra.2008.01.036 ◽

2008 ◽

Vol 319 (10) ◽

pp. 4199-4221 ◽

Cited By ~ 3

Author(s):

Heidi Haynal

Keyword(s):

Polynomial Rings ◽

Skew Polynomial Rings ◽

Skew Polynomial

Download Full-text

Skew polynomial rings of formal triangular matrix rings

Journal of Algebra ◽

10.1016/j.jalgebra.2011.10.014 ◽

2012 ◽

Vol 349 (1) ◽

pp. 201-216 ◽

Cited By ~ 3

Author(s):

Hoger Ghahramani

Keyword(s):

Triangular Matrix ◽

Polynomial Rings ◽

Matrix Rings ◽

Skew Polynomial Rings ◽

Skew Polynomial

Download Full-text

Partial Skew Polynomial Rings Over Semisimple Artinian Rings

Communications in Algebra ◽

10.1080/00927870902973888 ◽

2010 ◽

Vol 38 (5) ◽

pp. 1663-1676 ◽

Cited By ~ 1

Author(s):

Wagner Cortes ◽

Miguel Ferrero ◽

Yasuyuki Hirano ◽

Hidetoshi Marubayashi

Keyword(s):

Polynomial Rings ◽

Artinian Rings ◽

Skew Polynomial Rings ◽

Skew Polynomial

Download Full-text

ANNIHILATOR CONDITIONS IN MATRIX AND SKEW POLYNOMIAL RINGS

Journal of Algebra and Its Applications ◽

10.1142/s021949881250079x ◽

2012 ◽

Vol 11 (04) ◽

pp. 1250079 ◽

Cited By ~ 7

Author(s):

A. ALHEVAZ ◽

A. MOUSSAVI

Keyword(s):

Polynomial Rings ◽

Trivial Extension ◽

Skew Polynomial Ring ◽

Matrix Rings ◽

Skew Polynomial Rings ◽

Armendariz Rings ◽

Ore Extensions ◽

Necessary And Sufficient ◽

Skew Polynomial ◽

Armendariz Ring

Let R be a ring with an endomorphism α and α-derivation δ. By [A. R. Nasr-Isfahani and A. Moussavi, Ore extensions of skew Armendariz rings, Comm. Algebra 36(2) (2008) 508–522], a ring R is called a skew Armendariz ring, if for polynomials f(x) = a0 + a1 x + ⋯ + anxn, g(x) = b0+b1x + ⋯ + bmxm in R[x; α, δ], f(x)g(x) = 0 implies a0bj = 0 for each 0 ≤ j ≤ m. In this paper, radicals of the skew polynomial ring R[x; α, δ], in terms of a skew Armendariz ring R, is determined. We prove that several properties transfer between R and R[x; α, δ], in case R is an α-compatible skew Armendariz ring. We also identify some "relatively maximal" skew Armendariz subrings of matrix rings, and obtain a necessary and sufficient condition for a trivial extension to be skew Armendariz. Consequently, new families of non-reduced skew Armendariz rings are presented and several known results related to Armendariz rings and skew polynomial rings will be extended and unified.

Download Full-text