2016/11/11 by Sagar Shrivastava, Vaibhav Pandey, Shrivastava, Sagar +1
Mathematics · #Advanced Mathematical Theories #Algebraic and Geometric Analysis #Classical Analysis and ODEs (math.CA) #Commutative Algebra (math.AC) #FOS: Mathematics #Rings and Algebras (math.RA) #advanced mathematical theories #math.AC #math.CA #math.RA
paper · pdf · doi:10.48550/arxiv.1611.03667
openalex publication_date 2016/11/11 · arxiv created 2016/11/14 · arxiv updated 2016/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the ring of real analytic functions defined on [0,1], i.e. Cω[0,1] =\lbrace f :[0,1] \longrightarrow ℝ | f is analytic on [0,1]\rbrace In this article, we explore the nature of ideals in this ring. It is well known that the ring C[0,1] of real valued continuous functions on [0,1] has precisely the following maximal ideals: For γ∈ [0,1], Mγ := \lbrace f ∈ C[0,1] | f(γ) =0\rbrace It has been proved that each such Mγ is infinitely generated, in-fact uncountably generated. Observe that Cω[0,1] is a subring of C[0,1] We prove that for any γ in [0,1], the contraction Mωγ of Mγ under the natural inclusion of Cω[0,1] in C[0,1] is again a maximal ideal (of Cω[0,1] ), and these are precisely all the maximal ideals of Cω[0,1]. Next we prove that each Mωγ is principal (though Mγ is uncountably generated). Surprisingly, this forces all the ideals of the ring Cω[0,1] to be singly generated, i.e. Cω[0,1] is a PID.