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

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 $$\mathrm{diam}(Z(R)) = 2$$.

### 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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### References:

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