Germs and the local ring of a classical affine variety

For a point $x$ of an affine algebraic set $X$, a germ of a regular function at $x$ is a pair $(U,s)$, where $U$ is an open neighbourhood of $x$ and $s\in {\mathcal{O}}_X(U)$, modulo equality on some open neighbourhood of $x$ contained in both domains. Write ${\mathcal{O}}_{X,x}$ for the set of germs. Reflexivity uses $U$; symmetry reverses equality; transitivity intersects the two witness neighbourhoods, which still contain $x$. Add and multiply representatives after restricting to their intersection. If either representative is replaced by an equivalent one, intersect the two equality neighbourhoods: there both sums and both products agree. Thus the operations are well-defined. The restriction and algebra laws of Classical regular functions satisfy locality and unique gluing on a common finite intersection give a unital $k$-algebra; constants define its structure map. Evaluation of a germ at $x$ is well-defined by the same equality condition. Its local-ring property is proved in the following theorem.

Source comparison: Milne, Algebraic Geometry, v6.10, §3b, p. 60. Conventions here distinguish arbitrary affine algebraic sets from nonempty irreducible varieties.