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19 - M. Behboodi (with Z. Rakee), The annhilating-ideal graph of commutative rings II. J. Algebra Appl. 10 (4) (2011), 740-753. (ISI)

In this paper we continue our study of annihilating-ideal graph of commutative rings, that was introduced in (The annihilating-ideal graph of commutative rings I, to appear in J. Algebra Appl.). Let R be a commutative ring with A(R)be its set of ideals with nonzero annihilator and Z(R) its set of zero divisors. The annihilating-ideal graph of R is defined as the (undirected) graph AG(R) that its vertices are A(R)∗ = A(R)\{(0)} in which for every distinct vertices I and J, I− − −J is an edge if and only if IJ =(0). First, we study the diameter of AG(R). A complete characterization for the possible diameter is given exclusively in terms of the ideals of R when either R is a Noetherian ring or Z(R) is not an ideal of R. Next, we study coloring of annihilating-ideal graphs. Among other results, we characterize when either χ(AG(R)) ≤ 2or R is reduced and χ(AG(R)) ≤∞. Also it is shown that for each reduced ring R, χ(AG(R)) = cl(AG(R)). Moreover, if χ(AG(R)) is finite, then R has a finite number of minimal primes, and if n is this number, then χ(AG(R)) = cl(AG(R)) = n. Finally, we show that for a Noetherian ring R,cl(AG(R)) is finite if and only if for every ideal I of R with I2 =(0), I has finite number of R-submodules.

August, 2011


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