2010/09/22 by Ahmad Shafiei Deh Abad, Abad, Ahmad Shafiei Deh
Computer Science · Mathematics · #Commutative Algebra (math.AC) #FOS: Mathematics #Polynomial and algebraic computation #Rings and Algebras (math.RA) #math.AC #math.RA
paper · pdf · doi:10.48550/arxiv.1009.4280
38 pages
arxiv created 2010/09/22 · openalex publication_date 2010/09/22 · arxiv updated 2010/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper which is the first of a series of papers on smooth structures, the concepts of C-structures and smooth structures are introduced and studied. The notion of smooth structure on semi-integral domains is given. It is shown that each semi-integral domain which is not a field, admits a unique smooth structure and a large class of non-polynomial smooth functions on some semi-integral domains is constructed. A smooth function from Z-0 into Z is given which does not extend to a smooth function on Z. No concept from topology is used. As an application, it is shown that: Theorem - Let M and N be finite dimensional smooth manifolds. The algebra of real smooth functions on M (resp. N) will be denoted by A (resp. B). Assume that T is a homomorphism from B into A. Then, there exists exactly one smooth mapping f from M into N such that T=f*.