On the regular digraph of ideals of commutative rings

Authors

  • M. Afkhami
  • M. Karimi
  • K. Khashyarmanesh

Abstract

‎Let $R$ be a commutative ring‎. ‎The regular digraph of ideals of $R$‎, ‎denoted by‎ ‎$\Gamma(R)$‎, ‎is a digraph whose vertex-set is the set of all non-trivial ideals of $R$ and‎, ‎for every‎ ‎two distinct vertices $I$ and $J$‎, ‎there is an arc from $I$ to $J$ whenever $I$ contains‎ ‎a non-zero divisor on $J$‎. ‎In this paper‎, ‎we study the connectedness of $\Gamma(R)$‎. ‎We also completely characterize the diameter of this graph and determine the number of edges in $\Gamma(R)$‎, ‎whenever $R$ is a finite direct product of fields‎. ‎Among other things‎, ‎we prove that $R$ has finite number of ideals‎ ‎if and only if $\T{N}_{\Gamma(R)}(I)$ is finite‎, ‎for all vertices $I$ in $\Gamma(R)$‎, ‎where $\T{N}_{\Gamma(R)}(I)$ is‎ ‎the set of all adjacent vertices to $I$ in $\Gamma(R)$‎. 10.1017/S0004972712000792

Published

2013-08-10

Issue

Vol. 88 No. 2 (2013)

Section

Articles
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