Identities of sum of two PI-algebras in the case of positive characteristic

FAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOWe consider the following question posted by Beidar and Mikhalev in 1995 for an associative ring R = R1 + R2: is it true that if the subrings R1 and R2 satisfy polynomial identities, then R also satisfies a polynomial identity? Over a field of positive characteristic we establish new conditions on R1 and R2 that guarantee a positive answer to the question. We find upper and low bounds on the degrees of identities of R. © 2015 World Scientific Publishing Company.We consider the following question posted by Beidar and Mikhalev in 1995 for an associative ring R = R1 + R2: is it true that if the subrings R1 and R2 satisfy polynomial identities, then R also satisfies a polynomial identity? Over a field of positive characteristic we establish new conditions on R1 and R2 that guarantee a positive answer to the question. We find upper and low bounds on the degrees of identities of R.25812651273FAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOFAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO2013/15539-2Bahturin, Y., Giambruno, A., Identities of sums of commutative subalgebras (1994) Rend. Cir. Mat. Palermo, 43 (2), pp. 250-258Beidar, K.I., Mikhalev, A.V., Generalised polynomial identities and rings which are sums of two subrings (1995) Algebrai Logica, 34, pp. 3-11. , (in Russian)(1995) Algebra Logic, 34, pp. 1-5Bokut, L.A., Embeddings in simple associative algebras (1976) Algebra i Logica, 15, pp. 117-142. , (in Russian)(1976) Algebra Logic, 15, pp. 73-90Felzenszwalb, B., Giambruno, A., Leal, G., On rings which are sums of two PIsubrings: A combinatorial approach (2003) Pacific J. Math., 209 (1), pp. 17-31Goto, M., Note on a characterization of solvable Lie algebras (1962) J. Sci. Hiroshima Univ. Ser. A-I, 26, pp. 1-2Kegel, O.H., Zur hilpotenz gewisser assoziativer ringe (1962) Math. Ann., 149, pp. 258-260Kemer, A.R., The standard identity in charecteristic p: A conjecture of I.B. Volichenko (1993) Israel J. Math., 81, pp. 343-355Kepczyk, M., On algebras that are sums of two subalgebras satisfying certain polynomial identities (2008) Publ. Math. Debrecen, 72 (3), pp. 257-267Kepczyk, M., Note on algebras which are sums of two PI subalgebras (2015) J. Algebra Appl., 14 (10), p. 1550149Kepczyk, M., Puczylowski, E.R., On radicals of rings which are sums of two subrings (1996) Arch. Math., 66, pp. 8-12Kepczyk, M., Puczylowski, E.R., Rings which are sums of two subrings (1998) J. Pure Applied Algebra, 133, pp. 151-162Kepczyk, M., Puczylowski, E.R., Rings which are sums of two subrings satisfying a polynomial identity (2001) Commun. Algebra, 29, pp. 2059-2065Kolman, B., Semi-modular Lie algebras (1965) J. Sci. Hiroshima Univ. Ser. A-I, 29, pp. 149-163Panyukov, V.V., On the solvability of Lie algebras of positive characteristic (1990) Russ. Math. Surv., 45 (4), pp. 181-182Pchelintsev, S.V., Kegel theorem for alternative algebras (1985) Sibirsk. Mat. Zh., 26, pp. 195-196Petravchuk, A.P., Lie algebras which can be decomposed into the sum of an abelian subalgebra and a nilpotent subalgebra (1988) Ukrain. Math. J., 40, pp. 331-334Regev, A., Existence of polynomial identities in A F B (1971) Bull. Amer. Math. Soc., 77, pp. 1067-1069Regev, A., Existence of identities in A B (1972) Israel J. Math., 11, pp. 131-152Rowen, L.H., Generalized polynomial identities II (1976) J. Algebra, 38, pp. 380-392Salwa, A., Rings that are sums of two locally nilpotent subrings (1996) Commun. Algebra, 24 (12), pp. 3921-3931Zusmanovich, P., A Lie algebra that can be written as a sum of two nilpotent subalgebras is solvable (1991) Math. Notes, 50, pp. 909-91