For a (not necessarily unital) ring \(R\), an additive map \(\varphi\colon R\to R\) is called a left centralizer, if \(\varphi(xy)=\varphi(x)y\) for all \(x,y\in R\). The paper under review studies a more general condition, when the multiplication is replaced by an arbitrary multilinear polynomial in noncommuting variables. For an algebra \(A\) over a commutative ring \(\Phi\) with unity, the authors fix \(\alpha_\pi\in\Phi\), \(\pi\in S_n\), and require that \[ \varphi\Bigl(\sum_{\pi\in S_n}\alpha_\pi x_{\pi(1)}x_{\pi(2)}\cdots x_{\pi(n)}\Bigr)=\sum_{\pi\in S_n}\alpha_\pi\varphi(x_{\pi(1)})x_{\pi(2)}\cdots x_{\pi(n)}\quad\text{for all }x_1,\dots,x_n\in A. \] There are two main results in the paper. The first one describes such maps \(\varphi\) acting on Lie subalgebras of \(A\). The second one shows that if \(R\) is a prime ring with the property \(\varphi(x^n)=\varphi(x)x^{n-1}\), then, under natural restrictions on the characteristic, \(\varphi\) is a left centralizer. The proofs use essentially functional identities of rings and algebras.