The zero-divisor graph of an amalgamated algebra
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The zero‑divisor graph of an amalgamated algebra Y. Azimi1,2 · M. R. Doustimehr1,2 Received: 5 June 2020 / Accepted: 13 August 2020 © Springer-Verlag Italia S.r.l., part of Springer Nature 2020
Abstract Let R and S be commutative rings with identity, f ∶ R → S a ring homomorphism and J an ideal of S. Then the subring R ⋈f J ∶= {(r, f (r) + j) ∣ r ∈ R and j ∈ J} of R × S is called the amalgamation of R with S along J with respect to f. In this paper, we generalize and improve recent results on the computation of the diameter of the zero-divisor graph of amalgamated algebras and obtain new results. In particular, we provide new characterizations for completeness of the zero-divisor graph of amalgamated algebra, as well as, a complete description for the diameter of the zero-divisor graph of amalgamations in the special case of finite rings. Keywords Amalgamated algebra · Amalgamated duplication · Graph of zero-divisor · Trivial extension Mathematics Subject Classification Primary 05C25 · 13A15 · 13B99
1 Introduction The concept of the graph of the zero-divisors of a ring R was introduced by Beck in [8] and further studied by Anderson and Naseer in [3]. In their work, all elements of the ring were vertices of the graph. Then, in [2], Anderson and Livingston considered only the set of non-zero zero-divisors of R as vertices. The definition of Anderson and Livingston (which we adopt in this paper) is as follows. The zero-divisor graph of R, denoted by Γ(R) , is the undirected graph with vertices 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 . Recall that the distance between (connected) vertices x and y, denoted by d(x, y), is the length of a shortest path connecting them. The This research was in part supported by a Grant from IPM (No.991300116). * Y. Azimi [email protected] M. R. Doustimehr [email protected] 1
Department of Mathematics, University of Tabriz, Tabriz, Iran
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School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395‑5746, Tehran, Iran
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diameter of a connected graph G, denoted by diam(G) , is the supremum of the distances between vertices. By [2, Theorem 2.3], Γ(R) is connected and diam(Γ(R)) ≤ 3. D’Anna, Finocchiaro, and Fontana in [9] and [10] have introduced the following construction. Let R and S be two commutative rings with identity, let J be an ideal of S and let f ∶ R → S be a ring homomorphism. They introduced the following subring (with standard component-wise operations)
R ⋈f J ∶= {(r, f (r) + j) ∣ r ∈ R and j ∈ J} of R × S , called the amalgamated algebra (or amalgamation) of R with S along J with respect to f. This construction generalizes the amalgamated duplication of a ring along an ideal (introduced and studied in [12]). Moreover, several classical constructions such as Nagata’s idealization (cf. [20, page 2], the R + XS[X] and the R + XS[[X]] constructions can be studied as particular cases of this new construction
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