for a moduli space $m$ of stable sheaves over a k3 surface $x$, we propose a series of conjectural identities in the chow rings $ch_\ast (m\times x^l)$, $l\geq 1$, generalizing the classic beauville-voisin identity for a k3 surface. we emphasize consequences of the conjecture for the structure of the tautological subring $r_\ast (m)\subset ch_\ast (m)$. we prove the proposed identities when $m$ is the hilbert scheme of points on a k3 surface. this is joint work with l. flapan, a. marian and r. silversmith.