Characterizations of second-order differential operators

{Let $N, k$ be positive integers with $k\geq 2$, and $\Omega \subset \mathbb{R}^{N}$ be a domain.} By the well-known properties of the Laplacian and the gradient, we have $$\Delta(f\cdot g)(x)=g(x) \Delta f(x)+f(x) \Delta g(x)+2\langle \nabla f(x), \nabla g(x)\rangle$$ for all $f, g\in \mathscr{C}^{k}(\Omega, \mathbb{R})$. {Due to the results of H.~K\"{o}nig and V.~Milman, Operator relations characterizing derivatives. Birkh\"{a}user / Springer, Cham, 2018.,} the converse is also true, i.e. this operator equation characterizes the Laplacian and the gradient under some assumptions. Thus the main aim of this paper is to provide an extension of this result and to study the corresponding equation $$T(f\cdot g)= fT(g)+T(f)g+2B(A(f), A(g)) \qquad \left(f, g\in P\right),$$ where $Q$ and $R$ are commutative rings, $P$ is a subring of $Q$ and $T\colon P\to Q$ and $A\colon P\to R$ are additive, while $B\colon R\times R\to Q$ is a symmetric and bi-additive. Related identities with one function will also be considered.