Coloring of commutative rings.(English)Zbl 0654.13001

In this interesting paper the author associates a graph R with a commutative ring R, where elements x and y are adjacent if and only if $$xy=0$$. Because of the presence of 0 the graph R is connected and of small diameter. In other settings it may well be very useful to use $$R^*=R-\{0\}$$ as the set of vertices of the associated graph. Nevertheless, the author introduces a collection of algebraic notions derived from graph theoretic ones and a conjecture which could well lead to much further speculation, i.e., that for commutative rings R and their graphs R, the clique number equals the chromatic number. This fact would indicate that not all simple graphs are graphs of commutative rings (there is a theorem here at least).
The author classifies all rings of chromatic number at most 3. It is noted that the conjecture holds for chromatic numbers up to 5. Furthermore, the beginnings of a graph-related ideal theory are also established. The most important conclusion of the paper is the following: this paper only scratches the surface of the combinatorial properties of the zero divisors in commutative rings. Ultimately, the study of colorings (commutative rings of finite chromatic number) will involve graph theory. Rings have been associated with graphs before, and in meaningful ways. This looks like a nice experiment which has succeeded, with more results to be expected.

MSC:

 13A99 General commutative ring theory 05C15 Coloring of graphs and hypergraphs
Full Text:

References:

 [1] de Bruijn, N.G; Erdøs, P, A colour problem for infinite graphs and a problem in theory of relations, (), 371-373 · Zbl 0044.38203 [2] König, D, Theorie der endlichen und unendlichen graphen, (1936), Reprinted Chelsea, New York, 1950 · JFM 62.0654.05 [3] {\scH. B. Eilertsen}, Graphs of 0-semigroups, unpublished. · Zbl 0957.34078
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