The interest in the symmetric polynomial identity \[ P_n(x_1,\ldots,x_n)=\sum_{\sigma\in S_n}x_{\sigma(1)}\ldots x_{\sigma(n)} \] in the theory of PI-algebras originates from the fact that over a field of characteristic 0 this identity is equivalent to the nil identity and the Nagata-Higman theorem gives that the algebra is nilpotent. The recent result of \textit{A. R. Kemer} [Isr. J. Math. 81, No. 3, 343-355 (1993; Zbl 0795.16017)]\ gives that over a field of positive characteristic every PI-algebra satisfies some symmetric identity. The theorem of \textit{M. Domokos} [Arch. Math. 63, No. 5, 407-413 (1994; Zbl 0809.16022)]\ shows that if the algebra \(R\) satisfies the symmetric identity of degree \(k\), then the \(n\times n\) matrix algebra \(M_n(R)\) satisfies a similar identity of degree \(nk\). On the other hand, the main result of \textit{M. B. Smirnov, A. E. Zalesskij} [Algebra Anal. 5, No. 6, 116-120 (1993; Zbl 0832.16021)]\ is that for every abelian group \(G\) of order \(n\) the \(G\)-graded algebra \(R=\sum_{g\in G}R_g\) over a field of characteristic \(p>0\) satisfies \(P_{np}\equiv 0\), provided that the component \(R_1\) is commutative. In the paper under review the authors give a purely combinatorial proof of the following theorem. Let \(G\) be a cancellative semigroup with 1 and let the \(G\)-graded associative ring \(R=\sum_{g\in G}R_g\) have the property that the set \(H=\{g\in G\mid R_g\not=0\}\) consists of \(n\) elements. If the component \(R_1\) satisfies the polynomial identity \(P_k\equiv 0\), then the whole algebra \(R\) satisfies \(P_{nk}\equiv 0\). As a consequence the authors obtain new proofs of the above mentioned results of Domokos and Smirnov and Zalesskij.