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- Title
ON A PROBLEM OF A. V. GRISHIN.
- Authors
BEKH-OCHIR, C.; RANKIN, S. A.; Bokut, L. A.
- Abstract
In this paper, 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 $x_1^p, x_1^px_2^p, \ldots$ 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, L2 = L3. Moreover, if the characteristic of k does not divide d, we prove that $R_1^{(d)}$ is an ideal of k〈X〉0 and thus in particular, $R_1^{(d)} = R_2^{(d)}$. Finally, we show that over any field of characteristic p > 2, $R_1^{(d)}\ne R_2^{(d)}$ and L1 ≠ L2.
- Subjects
GRISHIN, A. V.; ASSOCIATIVE algebras; FREE algebras; MATHEMATICAL proofs; ALGEBRAIC fields; ALGEBRAIC spaces; INFINITY (Mathematics)
- Publication
Journal of Algebra & Its Applications, 2012, Vol 11, Issue 1, p1250020-1
- ISSN
0219-4988
- Publication type
Article
- DOI
10.1142/S0219498811005464