Irreducible Divisor Graphs II

Abstract

Let R be an integral domain, and let x ∈ R be a nonzero nonunit that can be written as a product of irreducibles. Coykendall and Maney (to appear), defined the irreducible divisor graph of x, denoted G(x), as follows. The vertices of G(x) are the nonassociate irreducible divisors of x (each from a pre-chosen coset of the form π U(R) for π ∈ R irreducible). Given distinct vertices y and z, we put an edge between y and z if and only if yz|x. Finally, if y ⁿ |x but y ⁿ⁺¹ ∤ x, then we put n − 1 loops on the vertex y.

In this article, inspired by the approach of the authors from Akhtar and Lee (to appear), we study G(x) using homology. A connection is found between H ₁ and the cycle space of G(x). We also characterize UFDs via these homology groups.

ACKNOWLEDGMENTS

The author would like to thank K. H. Rose and R. Moore for their work on Xy-pic. Both this article and Coykendall and Maney (to appear) would have been much more difficult to write had it not been for their efforts.

Notes

²The only significant differences in the arguments are due to the fact that the authors in Akhtar and Lee (to appear) use integer coefficients for their homology.

³It should be pointed out that in Diestel (2000), cycle spaces are only defined for finite graphs. However, if G is an infinite graph, then Proposition 4.3 still follows as long as we demand that 𝒞(G) is generated only by cycles of finite length. For considerations of infinite cycles and cycle spaces of infinite graphs, the reader is referred to Diestel (2005) and Diestel and Kühn (2004a); Diestel and Kühn (2004b).

⁴Of course, if t _i(n−2) = n for all 0 ≤ i ≤ r, then we may rearrange our subscripts on π₀, π₁,…, π _n so that π _n does not show up in each term of α.

Communicated by L. Ein.