Factorization in integral domains.

commutative algebra;integral domains;monoid domains;factorization

Ryan H Gipson

University:University of Louisville

Department:Mathematics

关键词: commutative algebra;    integral domains;    monoid domains;    factorization;   

Others  :  https://ir.library.louisville.edu/cgi/viewcontent.cgi?article=4171&context=etd

美国|英语

来源: The Universite of Louisville's Institutional Repository

【 摘 要 】

We investigate the atomicity and the AP property of the semigroup rings F[X; M], where F is a field, X is a variable and M is a submonoid of the additive monoid of nonnegative rational numbers. In this endeavor, we introduce the following notions: essential generators of M and elements of height (0, 0, 0, . . .) within a cancellative torsion-free monoid Γ. By considering the latter, we are able to determine the irreducibility of certain binomials of the form Xπ − 1, where π is of height (0, 0, 0, . . .), in the monoid domain. Finally, we will consider relations between the following notions: M has the gcd/lcm property, F[X; M] is AP, and M has no elements of height (0, 0, 0, . . .).