Yangjiang WEI, Gaohua TANG

The iteration digraphs of finite commutative rings

For a finite commutative ring $S$ (resp., a finite abelian group $S$) and a positive integer $k\geqslant2$, we construct an iteration digraph $G(S, k)$ whose vertex set is $S$ and for which there is a directed edge from $a\in S$ to $b\in S$ if $b=a^k$. We generalize some previous results of the iteration digraphs from the ring $\mathbb{Z}_n$ of integers modulo $n$ to finite commutative rings, and establish a necessary and sufficient condition for $G(S, k_1)$ and $G(S, k_2)$ to be isomorphic for any finite abelian group $S$.

Anahtar Kelimeler:

Iteration digraph, isomorphic component, isomorphic digraph

The iteration digraphs of finite commutative rings

Keywords:

Iteration digraph, isomorphic component, isomorphic digraph,

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Deng G, Yuan P. Isomorphic digraphs from powers modulo p. Czech Math J 2011; 61: 771–779.
Gilmer RW Jr. Finite rings having a cyclic multiplicative group of units. Am J Math 1963; 85: 447–452.
Lucheta C, Miller E, Reiter C. Digraphs from powers modulo p. Fibonacci Quant 1996; 34: 226–239.
Meemark Y, Wiroonsri N. The digraph of the k th power mapping of the quotient ring of polynomials over finite fields. Finite Fields Th Appl 2012; 18: 179–191.
Raghavendran R. Finite associative rings. Compos Math 1969; 21: 195–229.
Robert F. Discrete Iterations. Springer Series in Comput Math Vol 6. Berlin, Germany: Springer-Verlag, 1986.
Rogers TD. The graph of the square mapping on the prime fields. Discrete Math 1996; 148: 317–324.
Sha M. Digraphs from endomorphisms of finite cyclic groups. ArXiV 2010.
Somer L, Kˇr´ıˇzek M. On semiregular digraphs of the congruence x k ≡ y (mod n). Comment Math Univ Carolin 2007; 48: 41–58.
Somer L, Kˇr´ıˇzek M. On symmetric digraphs of the congruence x k ≡ y (mod n). Discrete Math 2009; 309: 1999–2009.
Somer L, Kˇr´ıˇzek M. The structure of digraphs associted with the congruence x k ≡ y (mod n). Czech Math J 2011; 61: 337–358.
Szalay L. A discrete iteration in number theory. BDTF Tud K ¨ozl 1992; 8: 71–91 (in Hungarian).
Wei YJ, Nan JZ, Tang GH. The cubic mapping graph for the ring of Gaussian integers modulo n. Czech Math J 2011; 61: 1023–1036.
Wei YJ, Nan JZ, Tang GH. Structure of cubic mapping graph for the ring of Gaussian integers modulo n. Czech Math J 2012; 62: 527–539.
Wei YJ, Tang GH, Su HD. The square mapping graphs of finite commutative rings. Algebr Colloq 2012; 19: 569–580.