Commutative ring theory: Regular rings

Basic Algebraic Geometry

Factorization in integral domains

A graph whose spectrum consists entirely of integers is called an integral graph. We present a survey of results on integral graphs and on the corresponding proof techniques.

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A survey on integral graphs

Let R be a commutative ring with identity, Z(R) its set of zero-divisors, and Nil(R) its ideal of nilpotent elements. The zero-divisor graph of R is Γ(R) = Z(R)\{0}, with distinct vertices x and y adjacent if and only if xy = 0. In this article, we study Γ(R) for rings R with nonzero zero-divisors which satisfy certain divisibility conditions between elements of R or comparability conditions between ideals or prime ideals of R. These rings include chained rings, rings R whose prime ideals contained in Z(R) are linearly ordered, and rings R such that {0} ≠ Nil(R) ⊆ zR for all z ∈ Z(R)\Nil(R).

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On the Zero-Divisor Graph of a Ring

We deal here with the existence of half-way nonzero divisors for com­ plete intersections and with related properties of their Hilbert functions.

http://www.researchgate.net/profile/Moshe_Roitman/publication/265682035_On_complete_intersections_and_their_Hilbert_functions/links/541c29ee0cf2218008c4f3c2.pdf

Moshe Roitman

Papers

On complete intersections and their hilbert functions

On t-linked overrings

Polynomial extensions of atomic domains

On Mori domains and commutative rings with CC⊥ I

On Nagata's theorem for the class group