On a problem of A. V. Grishin

2009/09/11 by C. Bekh-Ochir, Bekh-Ochir, C., S. A. Rankin +1
Mathematics · #16R10 #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA #msc:16R10

paper · pdf · doi:10.48550/arxiv.0909.2266

arxiv created 2009/10/07 · arxiv updated 2009/12/01

Abstract

In this note, we offer a short proof of V. V. Shchigolev's result that over any field k of characteristic p>2, the T-space generated by x₁^p,x₁^px₂^p,... is finitely based, which answered a question raised by A. V. Grishin. More precisely, we prove that for any field of any positive characteristic, R₂^(d)=R₃^(d) for every positive integer d, and that over an infinite field of characteristic p>2, L₂=L₃. Moreover, if the characteristic of k does not divide d, we prove that R₁^(d) is an ideal of k₀<X> and thus in particular, R₁^(d)=R₂^(d). Finally, we show that for any field of characteristic p>2, R₁^(d) is not equal to R₂^(d) and L₁ is not equal to L₂.

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