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We study the following open question: If a ring R is the sum of two subrings A and B both satisfying a polynomial identity, does R itself satisfy a polynomial identity? We give a positive answer to this question in case R satisfies a special mixed identity or (AB) k C A for some k > 1 or A or B is a Lie ideal. Our approach is based on a comparative analysis of the sequences of codimensions of the three rings and their asymptotics. As a reward we obtain a bound on the degree of a polynomial identity satisfied by R as a function of the degree of an identity satisfied by A and B.

On rings which are sums of two PI-subrings: a combinatorial approach