On a problem of A. V. Grishin
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 x1p,x1px2p,... 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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