Abstract
Let R be a commutative ring with identity, and let τ be a relation on the nonzero, non-unit elements of R. In this paper we generalize the defi- nitions of a factorization and a U-factorization via a relation τ and construct a variety of graphs based on these generalizations. These graphs are then examined in an effort to determine ring-theoretic properties.
Highlights
Throughout, R will denote a commutative ring with identity, and D will denote a domain
It is straightforward to show that ∼ and ≈ form equivalence relations in a commutative ring with zero-divisors, but ∼= need not satisfy reflexivity, and is not always an equivalence relation
Let R be a commutative ring with identity and let τ be a symmetric relation on R#
Summary
Throughout, R will denote a commutative ring with identity, and D will denote a domain. We define a τ -refinement of a τ -factorization λa1 · · · an to be a factorization of the form (λλ1 · · · λn) · b11 · · · b1m1 · b21 · · · b2m2 · · · bn1 · · · bnmn where ai = λibi1 · · · bimi is a τ -factorization for each i This is slightly different from the original definition in [4] where no unit factor was allowed in front of the refinement of the τ -factorization. This modification means it is not necessary for the relation to be both associate preserving and refinable to refine τ -factorizations in the above fashion.
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