In mathematics, in the subfield of ring theory, a ring R is a polynomial identity ring if there is, for some N > 0, an element P other than 0 of the free algebra, Z<X₁, X₂, ..., X_N>, over the ring of integers in N variables X₁, X₂, ..., X_N such that for all N-tuples r₁, r₂, ..., r_N taken from R it happens that