Zero-divisor graphs of twisted partial skew generalized power series rings

Purpose – The aim of this paper is to investigate the relationship between the ring structure of the twisted partial skew generalized power series ring R G ; ≤ ; Θ (cid:1) (cid:3) (cid:1) (cid:3) and the corresponding structure of its zero-divisor graph Γ R G ; ≤ ; Θ (cid:1) (cid:3) (cid:1) (cid:3) . Design/methodology/approach – The authors first introduce the history and motivation of this paper. Secondly, the authors give a brief exposition of twisted partial skew generalized power series ring, in addition to presenting some properties of such structure, for instance, a-rigid ring, a-compatible ring and (G,a)-McCoy ring. Finally, the study ’ s main results are stated and proved. Findings – The authors establish the relation between the diameter and girth of the zero-divisor graph of twistedpartialskewgeneralizedpower seriesring R G ; ≤ ; Θ (cid:1) (cid:3) (cid:1) (cid:3) andthe zero-divisorgraph ofthe groundring R . The authors also provide counterexamples to demonstrate that some conditions of the results are not redundant. As well the authors indicate that some conditions of recent results can be omitted. Originality/value – The results of the twisted partial skew generalized power series ring embrace a wide range of results of classical ring theoretic extensions, including Laurent (skew Laurent) polynomial ring, Laurent (skew Laurent) power series ring and group (skew group) ring and of course their partial skew versions.


Introduction
During the last decades, several papers have studied the relation between the algebraic structure of rings and their related graphs. Perhaps one of the first papers connecting the graphs to rings dates back to 1963 when R. Swan [1] gave an elegant proof to a well-known theorem by Amitsur and Levitzki [2] that "the ring of all n 3 n matrices M n R ð Þ over a commutative ring R satisfies the standard polynomial identity S 2n x ð Þ ¼ 0." Swan's proof is based completely on the use of graph theory. Also this connection was pointed out by C. Chao and M. Schutzenberger (see [3], p. 167]).
Recently, Beck's results on coloring of a commutative ring attract the interest of many mathematicians to explore the structure of rings through their zero-divisor graph [4]. Beck considered a commutative ring R as a simple graph whose vertices are all elements of R, such that two different vertices x, y ∈ R are adjacent if and only if xy 5 0. Beck's investigation of colorings was then continued by Anderson and Naseer in [5]. Anderson and Livingston redefined Beck's graph of a commutative ring R by restricting the vertices to be the set Z * R ð Þ that consists of nonzero zero divisors of R and called such graph a zero-divisor graph of R (see [6]). Thereafter, Redmond extended this concept to noncommutative case, and he gave two different ways to define zero-divisor graph of a noncommutative ring R. The first is directed and denoted by Γ R ð Þsuch that x → y is an edge between distinct vertices x and y if and only if xy 5 0. The second graph is undirected and denoted by Γ R ð Þsuch that two different vertices x and y are adjacent if and only if xy 5 0 or yx 5 0 (see [7,8]).
Afterward, many authors studied the relationship between zero-divisor graph of a ring R and zero-divisor graph of some of its extensions, for example, polynomial ring R x ½ , formal power series ring R x ½ ½ and skew generalized power series ring R S; σ ½ (see for example [9][10][11]). In this paper, we consider the (undirected) zero-divisor graph Γ R ð Þ of a ring R. For two distinct vertices x and y in Γ R ð Þthe distance between x and y, denoted by d(x, y), constitute the length of the shortest path connecting x and y, if such a path exists; otherwise d(x, y)d∞. Axtell et al. in [10] proved that if R is a commutative ring with identity and not isomorphic to According to [12], a twisted partial skew generalized power series ring R G;6 ; Θ Â Ã Â Ã embraces a wide range of classical ring theoretic extensions, including Laurent (skew Laurent) polynomial ring RCxD (RCx; σD), Laurent (skew Laurent) power series ring RCCxDD (RCCx; σDD) and group (skew group) ring R G ) and of course their partial skew versions. Our purpose of this paper is to continue study the relationship between zero-divisor graph of a ring R and zero-divisor graph of twisted partial skew generalized power series ring R G;6 ; Θ Â Ã Â Ã . In the following section, we give a brief exposition of twisted partial skew generalized power series ring R G;6 ; Θ Â Ã Â Ã , in addition to presenting some properties of such structure, for instance, α À rigid ring, α À compatible ring and G; α ð Þ− McCoy ring. In Section 3, our main results are stated and proved. We establish the relation between the diameter and girth of zero-divisor graph of twisted partial skew generalized power series ring R G;6 ; Θ Â Ã Â Ã and zero-divisor graph of the ground ring R. We also provide counterexamples to demonstrate that some conditions of the results are not redundant. As well we indicate that some conditions of recent results can be omitted.

Twisted partial skew generalized power series ring
The action of groups on sets is one of the crucial tools in study theory of representations and the algebraic structures of groups and rings. Partial action of groups on sets has been raised in functional analysis (see for instance [13,14]), then it was studied from a purely algebraic point of view. In [15], Dokuchaev and Exel defined partial skew group rings and proved that, under some assumptions, it is an associative ring. As a parallel case of partial skew group ring, Cortes and Ferrero defined partial skew polynomial rings and studied its prime and maximal ideals [16]. Thereafter many contemporaneous researchers were interested in studying the transfer of a lot of properties such as right Goldie, Baer, ACC on right annihilators, right p.p. and right zip properties between partial skew polynomial rings, partial skew Laurent polynomial rings and their ground ring (see for instance [17,18]). Fahmy et al. in [19] studied the transfer of right (left) zip property between the partial skew generalized power series ring R G;≤ ; α Â Ã Â Ã and its ground ring R. The twisted partial skew version was defined and studied in [12,20].
Let us first recall the definition of an idempotent (unital) twisted partial action, which is inspired by [ 1. An idempotent twisted partial action of a group G on a ring R is a triple ð Þ, the group of units of R, satisfying the following postulates, for all u, v and w in G: An ordered group G; $; 6 ð Þis called a strictly ordered group if it is satisfying the condition, if u, v, w ∈ G and u a v, then uw a vw and wu a wv. A subset X of G; $; 6 ð Þis said to be Artinian if every strictly decreasing sequence of elements of X is finite and that X is narrow if every subset of pairwise order-incomparable elements of X is finite.
The twisted partial skew generalized power series ring was introduced in [12, Definition 1.2] as follows.
Definition 2.2. Let R be a ring, G; 6 ð Þa strictly ordered group and Θ an idempotent twisted partial action of G on R. The twisted partial skew generalized power series ring gis Artinian and narrow subset of G, with pointwise addition, and the product operation is defined by According to Krempa [24], an endomorphism σ of a ring R is said to be rigid if aσ a ð Þ ¼ 0 implies a 5 0 for a ∈ R. If there exists a rigid endomorphism σ of R, then R is said to be σ À rigid. In [25], Hashemi and Moussavi generalized σ À rigid rings by introducing σ À compatible rings. A ring R is called σ À compatible if for each a, b ∈ R, ab 5 0 if and only if aσ(b) 5 0. If R is a ring, (S, 6) a strictly ordered monoid and ω: S → End(R) a monoid homomorphism, Marks et al. in [26] extended such concepts to S À compatible and S À rigid rings. A ring R is said to be S À compatible (respectively S À rigid) if ω s is compatible (respectively rigid) for every s ∈ S. The partial version of such concepts can be given as follows: Definition 2.3. Let R be a ring, G; 6 ð Þa strictly ordered group and Θ an idempotent twisted partial action of G on R. The ring R is called partial α À compatible if whenever s ∈ G, a ∈ D s , b ∈ R; ab 5 0 if and only if α s −1 a ð Þb ¼ 0.
According to Cortes [17] we adopt the following definition.
Definition 2.4. Let R be a ring, G; 6 ð Þa strictly ordered group and Θ an idempotent twisted partial action of G on R. The ring R is called partial α À rigid if a ∈ D s for s ∈ G such that α s −1 a ð Þa ¼ 0, then a 5 0.
A ring R is called Armendariz if whenever the polynomials f x ð Þ ¼ Σ m i¼0 a i x i and g x ð Þ ¼ Σ n j¼0 b j x j in the polynomial ring R x ½ , satisfy f x ð Þg x ð Þ ¼ 0 implies that a i b j 5 0 for all 0 ≤ i ≤ m and 0 ≤ j ≤ n. In [19], the partial skew version of Armendariz rings was defined as a natural extension of Definition 2 in [17]. In the light of the above definitions, we get the following: Definition 2.5. Let R be a ring, G; 6 ð Þa strictly ordered group and Θ an idempotent Recall that a monoid S (resp. a group G) is called a unique product monoid (u.p., for short) if for any two nonempty finite subsets X, Y ⊆ S (resp. G) there exist x ∈ X and y ∈ Y such that xy ≠ x 0 y 0 for every (x 0 , y 0 ) ∈ X 3 Y \{(x, y)}, the element xy is called a u.p. element of XY 5 {st: s ∈ X, t ∈ Y}. The class of u.p. monoids (resp. groups) includes the right and the left totally ordered monoids (resp. groups), for more details see [28].

Main results
In the following lemma, Z * l R ð Þ(Z * r R ð Þ) denote to the set of nonzero left (right) zero divisors of a ring R. and R is partial α À compatible, then f s ð Þ ∈ Z * l R ð Þ for some s ∈ supp f ð Þ. ii Then there exists a nonzero element g ∈ R G;6 ; Θ Â Ã Â Ã such that fg 5 0. Since G is a.n.u.p., there exist s ∈ supp f ð Þ and t ∈ supp g ð Þ such that st is u.p. of supp f ð Þ$supp g ð Þ.
Since R is partial α À compatible, it follows that f(s)g(t) 5 0. Hence f s ii Then there exists a nonzero element g ∈ R G;6 ; Θ Â Ã Â Ã such that gf 5 0. Since G is a.n.u.p., there exist t ∈ supp g ð Þ and s ∈ supp f ð Þ such that ts is u.p. of The following example shows that the partial α À compatibility condition for the ring R in part i ð Þ of the previous lemma is not superfluous.
Example 3.6. Let R ¼ Z 2 x ½ =Cx 2 þ xD and G; 6 ð Þ the group of integers Z with the trivial order. Let D 0 5 R, D 1 ¼ C1 þ xD, D −1 ¼ CxD and D i ¼ C0D for each i ∈ Zn 0; ±1 f g. Consider the identity automorphism α 0 of R and the isomorphism α 1 : D À1 → D 1 defined by α 1 x ð Þ ¼ 1 þ x. Then R G;6 ; Θ Â Ã Â Ã ≅ f P i a i y i j a i ∈ D i for each i ∈ Zg with pointwise addition, and the product operation is defined by a i y i À Á a j y j À Á ¼ α i ðα −i ða i Þa j Þy iþj : We conclude that and Γ R G;6 ; Θ Â Ã Â Ã is given by the following planar graph.
Before continuing, it worth mention here that accurate tracking of the proof of Proposition 3.21 in [11] shows that the condition on S to be a.n.u.p. is superfluous. Therefore, we can give the following: