1982/02/01 by D. D. Anderson, David F. Anderson · 1 voice · 1 citation
Mathematics · Medicine · #Rings, Modules, and Algebras #Magnolia and Illicium research #Spinal Hematomas and Complications #Mathematics #Section (typography) #Integral domain #Homogeneous #Domain (mathematical analysis) #Element (criminal law) #Divisibility rule #Pure mathematics #Graded ring #Algebra over a field #Mathematical analysis #Combinatorics #Law #Computer science
paper · pdf · doi:10.4153/cjm-1982-013-3
openalex publication_date 1982/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
Let R = ⊕ α ∊г R α be an integral domain graded by an arbitrary torsionless grading monoid Γ. In this paper we consider to what extent conditions on the homogeneous elements or ideals of R carry over to all elements or ideals of R. For example, in Section 3 we show that if each pair of nonzero homogeneous elements of R has a GCD, then R is a GCD-domain. This paper originated with the question of when a graded UFD (every homogeneous element is a product of principal primes) is a UFD. If R is Z + or Z -graded, it is known that a graded UFD is actually a UFD, while in general this is not the case. In Section 3 we consider graded GCD-domains, in Section 4 graded UFD's, in Section 5 graded Krull domains, and in Section 6 graded π-domains.