Let \(S_ n\) be the symmetric group on \(n\) letters and let \(\rho_ s=\sum_{t\in S_ n}(\text{sign}\;t)t^{-1}st\) for \(s\in S_ n\). The element \(\rho_ s\) depends only (up to a sign) on the conjugacy class of \(s\) and is non-zero if and only if the conjugacy class of \(s\) is determined by a partition with odd pairwise distinct parts. Let \(\lambda =(\lambda_ 1,\ldots,\lambda_ k)\), \(\lambda_ 1>\lambda_ 2>\ldots>\lambda_ k>0\), be such a partition of \(n\), and let \(K\) be any field of characteristic zero. The main result of the paper says that if \(s\) is an element of cycle-type \(\lambda\) in \(S_ n\), then \(\rho_ s\) belongs to the minimal two-sided ideal of the group ring \(K[S_ n]\) corresponding to the Young diagram whose hooks of boxes on the main diagonal are symmetric of lengths \(\lambda_ 1,\lambda_ 2,\ldots,\lambda_ k\), respectively. The proof given in the paper is in terms of the theory of matrix identities with trace developed by one of the authors in [Izv. Akad. Nauk SSSR, Ser. Mat. 38, 723--756 (1974; Zbl 0311.16016)]. However, there is a remark in a footnote that an alternative proof of the main theorem follows from the Frobenius results concerning representations of the alternating group.