Much work has been done on generalized factorization techniques in integral domains, namely τ -factorization. There has also been substantial progress made in investigating factorization in commutative rings with zero-divisors. This paper seeks to synthesize work done in these two areas and extend the notion of τ -factorization to commutative rings that need not be domains. In addition, we look into particular types of τ relations, which are interesting when there are zero-divisors present. We then proceed to classify commutative rings that satisfy the finite factorization properties given in this paper.

In this paper, we introduce a geometric statistic called the sprawl of a group with respect to a generating set, based on the average distance in the word metric between pairs of words of equal length. The sprawl quantifies a certain obstruction to hyperbolicity. Group presentations with maximum sprawl (ie without this obstruction) are called statistically hyperbolic. We first relate sprawl to curvature and show that nonelementary hyperbolic groups are statistically hyperbolic, then give some results for products and for certain solvable groups. In free abelian groups, the word metrics are asymptotic to norms induced by convex polytopes, causing several kinds of group invariants to reduce to problems in convex geometry. We present some calculations and conjectures concerning the values taken by the sprawl statistic for the group Z

Recently substantial progress has been made on generalized factorization techniques in integral domains, in particular τ -factorization. There has also been advances made in investigating factorization in commutative rings with zero-divisors. One approach which has been found to be very successful is that of U-factorization introduced by C.R. Fletcher. We seek to synthesize work done in these two areas by generalizing τ -factorization to rings with zero-divisors by using the notion of U-factorization.

Abstract. In this paper, we continue the program initiated by I. Beck's now classical paper concerning zero-divisor graphs of commutative rings. After the success of much research regarding zero-divisor graphs, many authors have turned their attention to studying divisor graphs of non-zero elements in the ring, the so called irreducible divisor graph. In this paper, we construct several different associated irreducible divisor graphs of a commutative ring with unity using various choices for the definition of irreducible and atomic in the literature. We continue pursuing the program of exploiting the interaction between algebraic structures and associated graphs to further our understanding of both objects. Factorization in rings with zero-divisors is considerably more complicated than integral domains; however, we find that many of the same techniques can be extended to rings with zero-divisors. This allows us to not only find graph theoretic characterizations of many of the finite factorization properties that commutative rings may possess, but also understand graph theoretic properties of graphs associated with certain commutative rings satisfying nice factorization properties. 2010

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