Simply, if there is a polynomial $f$, in noncommuting variables, which vanishes under substitutions from ring $R$, the ring will be called a PI ring (Polynomial Identity ring). For example, commutative rings always satisfy the polynomial $f(x,y) = xy - yx$.
Is $M_{2}(K)$, the ring of all $2 \times2$ matrices over a field $K$, a PI ring? I have tried to construct a polynomial, assuming that this ring is a PI ring, but couldn't find any. Thanks.