Consider the non-commutative polynomials \[ s_ n = \sum_{\pi \in \text{Sym}(n)} x_{\pi(1)} \dots x_{\pi(n)}\quad \text{and} \quad d_ n = \sum_{\pi \in \text{Sym}(n)} x_{\pi(1)} y_ 1 \dots y_{n - 1} x_{\pi(n)}. \] Let \(R\) be an algebra over a field of characteristic \(p > 0\). We show that if \(s_ n = 0\) (\(d_ n = 0\), resp.) is a polynomial identity on \(R,\) then \(s_{kn} = 0\), \((d_{(n - 1) k^ 2 + 1} = 0\), resp.) is an identity on the \(k \times k\) matrix algebra over \(R\). These results generalize an earlier theorem due to Chang and Zalesskij on the identities of the matrix algebra over a field of positive characteristic. Motivated by Regev's results on algebras (over a characteristic zero field) satisfying a Capelli identity we investigate what can be said about the modular cocharacter sequence of \(R\) if we assume that \(R\) satisfies the identity \(d_ n = 0\), and prove that certain types of irreducible \(\text{Sym}(n)\)-modules can not occur in the modular cocharacter.