On a problem of A. V. Grishin

TL;DR

This paper provides a concise proof of a key result in algebraic structures, showing finite basis properties of certain T-spaces over fields of positive characteristic, and clarifies their relationships under various conditions.

Contribution

It offers a short proof of V. V. Shchigolev's result on finitely based T-spaces and explores their structural relationships over different fields and characteristics.

Findings

R_2^{(d)}=R_3^{(d)} for all positive integers d over any field

L_2=L_3 over an infinite field of characteristic p>2

R_1^{(d)} is an ideal of k_0<X> when the characteristic does not divide d

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 x_1^p,x_1^px_2^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, R_2^{(d)}=R_3^{(d)} for every positive integer d, and that over an infinite field of characteristic p>2, L_2=L_3. Moreover, if the characteristic of k does not divide d, we prove that R_1^{(d)} is an ideal of k_0<X> and thus in particular, R_1^{(d)}=R_2^{(d)}. Finally, we show that for any field of characteristic p>2, R_1^{(d)} is not equal to R_2^{(d)} and L_1 is not equal to L_2.