## 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
Full Text:

### References:

 [1] DOI: 10.1016/S0021-8693(03)00370-3 · Zbl 1032.13014 [2] DOI: 10.1016/j.dam.2012.01.011 · Zbl 1239.05088 [3] Anderson D. F., Houston J. Math. 34 pp 361– (2008) [4] Anderson D. F., Comm. Algebra (2008) [5] DOI: 10.1016/j.jalgebra.2008.06.028 · Zbl 1158.13001 [6] Anderson D. F., J. Algebra Appl. (2012) [7] Anderson D. F., J. Algebra Appl. (2013) [8] DOI: 10.1016/j.jpaa.2011.12.002 · Zbl 1254.13003 [9] DOI: 10.1006/jabr.1998.7840 · Zbl 0941.05062 [10] DOI: 10.1016/j.jpaa.2006.10.007 · Zbl 1119.13005 [11] DOI: 10.1016/S0022-4049(02)00250-5 · Zbl 1076.13001 [12] DOI: 10.1007/978-1-4419-6990-3_2 · Zbl 1225.13002 [13] DOI: 10.1016/0021-8693(88)90202-5 · Zbl 0654.13001 [14] DOI: 10.1142/S0219498811004896 · Zbl 1276.13002 [15] DOI: 10.1142/S0219498811004902 · Zbl 1276.13003 [16] DOI: 10.1007/978-1-4612-0619-4 [17] DOI: 10.1080/00927872.2011.638353 · Zbl 1285.13011 [18] DOI: 10.1142/S1793830911001309 · Zbl 1247.05098 [19] Tamizh Chelvam T., J. Algebra Appl. (2013) [20] Tamizh Chelvam T., J. Algebra Appl. (2013) [21] DOI: 10.1016/j.jalgebra.2007.10.015 · Zbl 1143.13029 [22] Huckaba J. A., Commutative Rings with Zero Divisors (1988) · Zbl 0637.13001 [23] DOI: 10.1007/s13369-011-0096-y [24] DOI: 10.1016/j.jalgebra.2006.01.019 · Zbl 1109.13006 [25] DOI: 10.1016/j.jalgebra.2006.01.057 · Zbl 1106.13029 [26] DOI: 10.1016/j.jalgebra.2008.09.040 · Zbl 1163.13003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.