Homology of zero-divisors

Let R be a commutative ring with unity. The set Z(R) of zero-divisors in a ring does not possess any obvious algebraic structure; consequently, the study of this set has often involved techniques and ideas from outside algebra. Several recent attempts, among them [2], [3] have focused on studying the so-called zero-divisor graph ΓR, whose vertices are the zero-divisors of R, with xy being an edge if and only if xy = 0. This object ΓR is somewhat unwieldy in that it has many symmetries; for example, if u ∈ R∗ is any unit, then x 7→ ux induces a (graph) automorphism of ΓR. One way of treating this issue – following an idea of Lauve [5] – is to work with the ideal zero-divisor graph IR. In effect, one replaces zero-divisors of R by proper ideals with nonzero annihilator; this is the approach adopted by the authors in [1]. Such a perspective also has its shortcomings; for instance, it does not adequately detect the phenomenon of there being three distinct proper ideals I, J,K in R with IJK = 0, but IJ 6= 0, IK 6= 0, JK 6= 0.