Let $V$ be an affine variety in $\mathbb{C}^{n}$, i.e. $V$ is the vanishing set of an ideal $I \subset \mathbb{C}[x_{1}, \dots, x_{n}]$. Furthermore let $g \in \mathbb{C}[x_{1}, \dots, x_{n}]$. Introduce the relation $\sim$ on $\mathbb{C}[x_{1},\dots, x_{n}]$, where $a \sim b \text{ if } g(a) = g(b)$ and consider the quotient $Q = V/\sim$. Likewise, the relation could be introduced for a finite set of polynomials $g_{1},\dots, g_{m}$.
I am looking for a reference on this kind of object. As far as I know (from googling around), in algebraic geometry usually only quotients of varieties by group actions are studied, and these quotients are then called quotient varieties, so another name for the above object should be used.