Functional Degrees And Arithmetic Applications, I: The Set Of Functional Degrees

2022/01/08 by P. L. Clark, Clark, P. L., U. Schauz +1
Mathematics · #13F20 #20C05 #20K01 #Advanced Topics in Algebra #Commutative Algebra (math.AC) #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Rings, Modules, and Algebras #math.AC #math.GR #msc:13F20 #msc:20C05 #msc:20K01

paper · pdf · doi:10.48550/arxiv.2201.02763

21 pages

arxiv created 2022/01/08 · openalex publication_date 2022/01/08 · arxiv updated 2022/01/11 · openalex created_date 2022/12/24 · openalex updated_date 2026/07/28

Abstract

We give a further development of the Aichinger-Moosbauer calculus of functional degrees of maps between commutative groups. For any fixed given commutative groups A and B, we compute the largest possible finite functional degree that a map f: A \longrightarrow B can have. We also determine the set of all possible degrees of such maps. This also yields a solution to Aichinger and Moosbauer's problem of finding the nilpotency index of the augmentation ideal of group rings of the form Z_p^β[Z_p^α1× Z_p^α2×\dotsm× Z_p^αn] with p,β,n,α₁,\dotsc,α_n∈ℤ⁺, p prime.

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