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The zero-divisor graph of an amalgamated algebra. (English) Zbl 1478.13010

In the paper under review, the authors improve some results on the completeness and diameter of the zero-divisor graph of amalgamated algebras. The zero-divisor graph of a commutative ring \(R\), denoted by \(\Gamma(R)\), is the undirected graph with vertices \(Z(R)^*=Z(R)\setminus\{0\}\), and for distinct \(x,y\in Z(R)^*\), the vertices \(x\) and \(y\) are adjacent if and only if \(xy = 0\).
Let \(R\) and \(S\) be commutative rings with identity, \(f:R\to S\) a ring homomorphism and \(J\) an ideal of \(S\). Then the subring \(R\bowtie^fJ:=\{(r,f(r)+j)\mid r\in R\) and \(j\in J\}\) of \(R\times S\) is called the amalgamation of \(R\) with \(S\) along \(J\) with respect to \(f\). It is known in [Y. Azimi et al., Commun. Algebra 47, No. 5, 2251–2261 (2019; Zbl 1426.13011)] that \(Z(R\bowtie^fJ)\subseteq Z_1\cup Z_2\) where \(Z_1=\{(r,f(r)+j)\mid r\in Z(R)\}\) and \(Z_2=\{(r,f(r)+j)\mid j'(f(r)+j)=0\) for some \(j'\in J\setminus\{0\}\}\). The amalgamated ring \(R\bowtie^fJ\) is said to have the condition \((\star)\) if the equality \(Z(R\bowtie^fJ)=Z_1\cup Z_2\) holds. Consider the following properties:
(a)
\(Z(R)^2=0\).
(b)
For every \(r\in R\setminus Z(R)\), and every \(0\neq j\in J\), \(jf(r)\neq0\).
(c)
For every \(0\neq r\in Z(R)\), and every \(0\neq j\in J\), \(jf(r)=0\).
(d)
\(J^2=0\).
It is shown that if \(\Gamma(R)\neq\emptyset\) and that \(R\bowtie^fJ\) has the condition \((\star)\), then \(\Gamma(R\bowtie^fJ)\) is complete; if and only if properties (a), (b), (c), and (d) holds; if and only if \(Z(R\bowtie^fJ)^2=0\). By examples the authors showed that it is necessary to assume that \(R\) is not a domain and \(R\bowtie^fJ\) has the condition \((\star)\).
Then the authors showed that when diameter of \(\Gamma(R\bowtie^fJ)\) is 2 or 3. Among other things, it is shown that if \(J\subseteq Nil(S)\) and if \(\Gamma(R\bowtie^fJ)\) satisfies properties (b) and (c) but not property \((a)\), then diam\((\Gamma(R\bowtie^fJ))=2\). By an example they showed that the condition \(J\subseteq Nil(S)\) is necessary.
The final section deals with finite ring case. The paper ends with several examples.

MSC:

13A70 General commutative ring theory and combinatorics (zero-divisor graphs, annihilating-ideal graphs, etc.)
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
13A15 Ideals and multiplicative ideal theory in commutative rings
13B99 Commutative ring extensions and related topics

Citations:

Zbl 1426.13011
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Full Text: DOI

References:

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