${\mathcal{I}}$ is a differential ideal we also have $\{{\mathcal{V}}_\lambda{\mathcal{I}}\}\subset{\mathcal{I}}[\lambda]$). Given a PVA ideal ${\mathcal{I}}$, we can consider the induced Poisson vertex algebra structure over the differential algebra ${\ensuremath{\raisebox{.40ex}{\ensuremath{{\mathcal{V}}}}
\! \big / \! \raisebox{-.40ex}{\ensuremath{{\mathcal{I}}}}}}$, which is called the *quotient PVA*.

In this paper we consider PVA structures on the algebra $R_I={\mathbb{F}}[u_i^{(n)}\mid i\in I,n\in{\mathbb{Z}}_+]$ of differential polynomials in the variables $\{u_i\}_{i\in I}$, where $I$ is an index set (possibly infinite). The derivation $\partial$ is defined by $\partial(u_i^{(n)})=u_i^{(n+1)}$, $i\in I,n\in{\mathbb{Z}}_+$. Note that on $R_I$ we have the following commutation relations: $\left[\frac{\partial}{\partial u_i^{(n)}},\partial\right]=\frac{\partial}{\partial u_i^{(n-1)}}$, where the RHS is considered to be zero if $n=0$.

\[master\] Let ${\mathcal{V}}=R_I$ and $H=\big(H_{ij}(\lambda)\big)_{i,j\in I}\in\operatorname{Mat}_{I\times I}{\mathcal{V}}[\lambda]$.

(a) There is a unique $\lambda$-bracket $\{\cdot\,_\lambda\,\cdot\}_H$ on ${\mathcal{V}}$, such that $\{u_i{}_\lambda u_j\}_H=H_{ji}(\lambda)$ for every $i,j\in I$, and it is given by the following Master Formula $$\label{masterformula}
    \{f_\lambda g\}_H=
    \sum_{\substack{i,j\in I\\m,n\in{\mathbb{Z}}_+}}\frac{\partial g}{\partial u_j^{(n)}}(\lambda+\partial)^n
    H_{ji}(\lambda+\partial)(-\lambda-\partial)^m\frac{\partial f}{\partial u_i^{(m)}}\,.
    $$

(b) The $\lambda$-bracket on ${\mathcal{V}}$ is skew-symmetric if and only if skew-symmetry holds on generators ($i,j\in I$): $$\label{skewsimgen}
    \{u_i{}_\lambda u_j\}_H=-\{u_j{}_{-\lambda-\partial}u_i\}_H\,,$$ and this is equivalent to skewadjointness of $H$.

(c) Assuming that the skew-symmetry condition holds, the $\lambda$-bracket satisfies the Jacobi identity, thus making ${\mathcal{V}}$ a PVA, provided that the Jacobi identity holds on any triple of 