On the dot product graph of a commutative ring. (English) Zbl 1316.13005

Badawi, Ayman

Commun. Algebra 43, No. 1, 43-50 (2015).

Let \(A\) be a commutative ring with nonzero identity, \(1 \leq n < \infty\) be an integer, and let \(R = A \times A \times\cdots \times A\) (\(n\) times). Let \(x = (x_1,\cdots, x_n)\), \(y = (y_1, \cdots, y_n)\in R\). Then the dot product \(x \cdot y = x_1y_1 + x_2y_2 + \cdots + x_ny_n\in A\). In the paper under review, the author introduces the total dot product graph of \(R\) to be the (undirected) graph \(TD(R)\) with vertices \(R^* = R \;\{(0,\cdots, 0)\}\), and two distinct vertices \(x\) and \(y\) are adjacent if and only if \(x \cdot y = 0\in A\). Let \(Z(R)\) denote the set of all zero-divisors of \(R\). Then the zero-divisor dot product graph of \(R\) is the induced subgraph \(Z(R)\) of \(TD(R)\) with vertices \(Z(R)^* = Z(R) \;\{(0,\cdots, 0)\}\). It follows that each edge (path) of the classical zero-divisor graph \(\Gamma(R)\) is an edge (path) of \(Z(R)\).
Among the other results, the author shows the following two results:
{ Theorem 1.} Let \(A\) be a commutative ring with \(1\neq 0\) that is not an integral domain, and let \(R = A \times A\). Then the following statements hold.

(1)

\(TD(R)\) is connected and \(\mathrm{diam}(TD(R)) = 3\).

(2)

\(Z(R)\) is connected, \(Z(R)\neq \Gamma(R)\), and \(\mathrm{diam}(Z(R)) = 3\).

(3)

\(\mathrm{girth}(Z(R)) = \mathrm{girth}(TD(R)) = 3\).

{Theorem 2.} Let \(A\) be a commutative ring with \(1\neq 0\). Then the following statements hold.

(1)

If \(A\) is an integral domain and \(R = A \times A \times A\), then \(Z(R)\) is connected and \(\mathrm{diam}(Z(R)) = 3\).

(2)

If \(A\) is not an integral domain and \(R = A \times A \times A\), then \(Z(R)\) is connected and \(\mathrm{diam}(Z(R)) = 2\).

(3)

If \(4 \leq n < \infty\) and \(R = A \times A \times\cdots\times A\) (\(n\) times), then \(Z(R)\) is connected and \(\mathrm{diam}(Z(R)) = 2\).

Reviewer: Siamak Yassemi (Tehran)

MSC:

13A15	Ideals and multiplicative ideal theory in commutative rings
13B99	Commutative ring extensions and related topics
05C99	Graph theory

Keywords:

annihilator graph; total graph; zero-divisor graph

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