Let R be an associative algebra over a field K generated by a vector subspace V. The polynomial f(x 1, . . . , xn ) of the free associative algebra K〈x 1, x 2, . . .〉 is a weak polynomial identity for the pair (R, V) if it vanishes in R when evaluated on V. We survey results on weak polynomial identities and on their applications to polynomial identities and central polynomials of associative and close to them nonassociative algebras and on the finite basis problem. We also present results on weak polynomial identities of degree three.

• 16R10 - \(T\)-ideals, identities, varieties of associative rings and algebras
• 16R30 - Trace rings and invariant theory (associative rings and algebras)
• 17B01 - Identities, free Lie (super)algebras
• 17B20 - Simple, semisimple, reductive (super)algebras
• 17C05 - Identities and free Jordan structures
• 17C20 - Simple, semisimple Jordan algebras
• 20C30 - Representations of finite symmetric groups

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