I'm confused on the definition of this sheaf of rings (taken from wikipedia):
Choose an open subset $U$ of some complex affine space $\mathbb{C}^n$, and fix finitely many holomorphic functions $f_1, ... , f_k$ in $U$. Let $X = V(f_1, ... , f_k)$ be the common vanishing locus of these holomorphic functions, that is, $X = \{x \in U : f_1(x) = \cdots = f_k(x) = 0\}$. Define a sheaf of rings on $X$ by letting $\mathcal O_X$ be the restriction to $X$ of $\mathcal O_U/(f_1, ... , f_k)$, where $\mathcal O_U$ is the sheaf of holomorphic functions on $U$. Then the locally ringed $\mathbb{C}$-space $(X, \mathcal O_X)$ is a local model space.
What is meant by $\mathcal O_U/(f_1, ... , f_k)$? Are we saying this (below)?
Define a sheaf of ideals $\mathscr I$ of $\mathcal O_U$ by defining $\mathscr I(W)$ to be the ideal of $\mathcal O_U(W)$ generated by $f_1|W, ... , f_k|W$ for $W$ open in $U$, then set $\mathcal O_U/\mathscr I$ to be the sheaf associated to the presheaf $W \mapsto \mathcal O_X(W)/\mathscr I(W)$. Then set $\mathcal O_X$ to be the inverse image sheaf of $\mathcal O_X/\mathscr I$ under the inclusion map $j: X \rightarrow U$.