The following question may be simple, but I tend to have bad intuition on the subject of algebra, so here goes:
Given a coordinate ring $R:=K[x_1,...,x_n]/\big( p_1,...,p_r \big)$ where $p_1,...,p_r \in K[x_1,...,x_n]$, and a polynomial $q\in K[x_1,...,x_n]$, is it true that:
$R/\overline{q}\cong K[x_1,...,x_n]/\big( p_1,...,p_r, q \big)$
I think this statement is true by the correspondence theorem for ideals, but could not coherently state an argument to myself proving as much. I would also appreciate any counter-examples, if the original statement is incorrect.
I'm writing the answer that Parthiv Basu wrote in the comments, so that I can tick this thread as closed (If my arguments are written incorrectly, feel free to edit):
Given a commutative ring $A$ with an ideal $I$, we know that for any ideal $J\supseteq I$ we have:
$A/J\cong \big(A/I \big)/ \big( J+I/ I \big)$ by the correspondece theorem for ideals.
In this case, for $A=K[x_1,...,x_n]$, $I=(p_1,...,p_r)$ and $J=(p_1,...,p_r,q)$, we have that $J+I=(\overline{q})$, and therefore:
$K[x_1,...,x_n]/J\cong R/(\overline{q})$