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Homology of zerodivisors Reza Akhtar and Lucas Lee

Summary: Homology of zero­divisors
Reza Akhtar and Lucas Lee
Let R be a commutative ring with unity. We define a semisimplicial abelian
group based on the structure of the semigroup of ideals of R and investigate
various properties of the homology groups of the associated chain complex.
1 Introduction
Let R be a commutative ring with unity. The set Z(R) of zero­divisors 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 so­called zero­divisor graph #R ,
whose vertices are the zero­divisors of R, with xy being an edge if and only if xy = 0.
This object #R is somewhat unwieldy in that it has many symmetries; for example,
if u # R # is any unit, then x ## ux induces a (graph) automorphism of #R . One
way of treating this issue -- following an idea of Lauve [5] -- is to work with the
ideal zero­divisor graph IR . In e#ect, one replaces zero­divisors 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 #= 0, IK #= 0, JK #= 0.


Source: Akhtar, Reza - Department of Mathematics and Statistics, Miami University (Ohio)


Collections: Mathematics