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

We consider the following question posted by K.I. Beidar and A.V. Mikhalev for a ring R=A+B: is it true that if subrings A and B satisfy polynomial identities, then R also satisfies a polynomial identity?

[1]  A. Petravchuk.  Lie algebras, decomposable into a sum of an Abelian and a nilpotent subalgebra , 1988 .

[2]  M. Kepczyk.  Note on algebras which are sums of two PI subalgebras , 2015 .

[3]  M. Kepczyk.  On algebras that are sums of two subalgebras satisfying certain polynomial identities , 2008, Publicationes Mathematicae Debrecen.

[4]  A. Mikhalev,et al.  Generalized polynomial identities and rings which are sums of two subrings , 1995 .

[5]  L. Rowen.  General polynomial identities. II , 1976 .

[6]  L. Rowen.  Generalized polynomial identities , 1975 .

[7]  E. Puczyłowski,et al.  RINGS WHICH ARE SUMS OF TWO SUBRINGS SATISFYING POLYNOMIAL IDENTITIES , 2001 .

[8]  E. Puczyłowski,et al.  On radicals of rings which are sums of two subrings , 1996 .

[9]  Y. Bahturin,et al.  Identities of sums of commutative subalgebras , 1994 .

[10]  E. Puczyłowski,et al.  Rings which are sums of two subrings , 1998 .

[11]  A. Regev.  Existence of identities inA⊗B , 1972 .

[12]  V. Panyukov.  COMMUNICATIONS OF THE MOSCOW MATHEMATICAL SOCIETY: On the solubility of Lie algebras of positive characteristic , 1990 .

[13]  Morikuni Gotô.  Note on a characterization of solvable Lie algebras. , 1962 .

[14]  A. Salwa.  Rings that are sums of two locally nilpotent subrings, II , 1996 .

[15]  G. Leal,et al.  On rings which are sums of two PI-subrings: A combinatorial approach , 2003 .

[16]  O. Kegel.  Zur Nilpotenz gewisser assoziativer Ringe , 1963 .

[17]  P. Zusmanovich.  A Lie algebra that can be written as a sum of two nilpotent subalgebras is solvable , 1991, 0911.5418.