For closed subschemes \(X_1, X_2 \subseteq Y\), their intersection is \(\fiberprod{X_1}{Y}{X_2}\), again a closed subscheme of \(Y\). For \(Y = \Spec R\) with \(X_i = \Spec(R/J_i)\) this is \[ \Spec\qty{ R/J_1 \tensor_R R/J_2 } \cong \Spec R/(J_1 + J_2) , \] so intersecting subschemes adds ideals.
On affine schemes, scheme-theoretic intersection is computed by adding the defining ideals.
Intersecting \(y = x^2\) with \(y = 0\) in \(\AA^2\) gives \(k[x,y]/(y - x^2, y) \cong k[x]/(x^2)\), the double point: tangency shows up as a length, and Bézout’s theorem counts lengths rather than points. Intersecting \(y = x^2\) with \(y = 1\) gives \(k[x]/(x^2-1)\), two reduced points, and the total length is \(2\) either way.
Note that the ideal sum need not be radical even when both \(J_i\) are, which is exactly why the reduced induced structure of the intersection is the wrong object for counting.