  We consider the following question posted by K.I. Beidar and A.V. Mikhalev in
1995 for an associative ring $R=R_1+R_2$: is it true that if the subrings $R_1$
and $R_2$ satisfy polynomial identities, then $R$ also satisfies a polynomial
identity? Over a field of positive characteristic we establish new conditions
on $R_1$ and $R_2$ that guarantee a positive answer to the question. We find
upper and low bounds on the degrees of identities of $R$.
