# Article

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Keywords:
iteration digraph; fundamental constituent; digraphs product
Summary:
For a finite commutative ring $R$ and a positive integer $k\geqslant 2$, we construct an iteration digraph $G(R, k)$ whose vertex set is $R$ and for which there is a directed edge from $a\in R$ to $b\in R$ if $b=a^k$. Let $R=R_1\oplus \ldots \oplus R_s$, where $s>1$ and $R_i$ is a finite commutative local ring for $i\in \{1, \ldots , s\}$. Let $N$ be a subset of $\{R_1, \dots , R_s\}$ (it is possible that $N$ is the empty set $\emptyset$). We define the fundamental constituents $G_N^*(R, k)$ of $G(R, k)$ induced by the vertices which are of the form $\{(a_1, \dots , a_s)\in R\colon a_i\in {\rm D}(R_i)$ if $R_i\in N$, otherwise $a_i\in {\rm U}(R_i), i=1,\ldots ,s\},$ where U$(R)$ denotes the unit group of $R$ and D$(R)$ denotes the zero-divisor set of $R$. We investigate the structure of $G_N^*(R, k)$ and state some conditions for the trees attached to cycle vertices in distinct fundamental constituents to be isomorphic.
References:
[1] Bini, G., Flamini, F.: Finite Commutative Rings and Their Applications. The Kluwer International Series in Engineering and Computer Science 680 Kluwer Academic Publishers, Dordrecht (2002). MR 1919698 | Zbl 1095.13032
[2] R. W. Gilmer, Jr.: Finite rings having a cyclic multiplicative group of units. Am. J. Math. 85 (1963), 447-452. DOI 10.2307/2373134 | MR 0154884 | Zbl 0113.26501
[3] Lucheta, C., Miller, E., Reiter, C.: Digraphs from powers modulo $p$. Fibonacci Q. 34 (1996), 226-239. MR 1390409 | Zbl 0855.05067
[4] Raghavendran, R.: Finite associative rings. Compos. Math. 21 (1969), 195-229. MR 0246905 | Zbl 0179.33602
[5] Somer, L., Křížek, M.: On semiregular digraphs of the congruence $x^k\equiv y\pmod n$. Commentat. Math. Univ. Carol. 48 (2007), 41-58. MR 2338828
[6] Somer, L., Křížek, M.: The structure of digraphs associated with the congruence $x^k\equiv y\pmod n$. Czech. Math. J. 61 (2011), 337-358. DOI 10.1007/s10587-011-0079-x | MR 2905408 | Zbl 1249.11006
[7] Wei, Y., Tang, G., Su, H.: The square mapping graphs of finite commutative rings. Algebra Colloq. 19 (2012), 569-580. MR 3073410 | Zbl 1260.13032
[8] Wilson, B.: Power digraphs modulo $n$. Fibonacci Q. 36 (1998), 229-239. MR 1627384 | Zbl 0936.05049

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