Pulling back the equation of a Weil divisor can give zero

Given: A field $k$, the affine line $Y={\mathbb{A}}_k^1=\mathrm{Spec}k[t]$ with origin $Z=V(t)=\{0\}$, the one-point scheme $X=\mathrm{Spec}k$, and the morphism $i:X\to Y$ corresponding to the $k$-algebra homomorphism $k[t]\to k$ with $t\mapsto 0$.

$Y=\mathrm{Spec}k[t]$ is a normal Noetherian integral scheme of dimension one; its prime divisors are the closed points $V(g)$ for the irreducible polynomials $g$, and $Z=V(t)$ is a prime divisor with local equation $t$. The element $t$ is a meromorphic unit on $Y$, that is, $t\in K(Y{)}^{\times }=k(t{)}^{\times }$, and it is a regular section of ${\mathcal{O}}_Y$, i.e. a nonzerodivisor in every local ring of $Y$ (Weil divisor normal noetherian scheme).

$X=\mathrm{Spec}k$ is a zero-dimensional integral scheme with ${\mathcal{O}}_X(X)=k$ and ${\mathcal{O}}_X=K_X$ the constant sheaf $k$ (Sheaf total quotient rings); it has no prime divisors, because a prime divisor requires local-ring dimension one at its generic point, whereas the unique stalk of $X$ is the zero-dimensional field $k$, so $\mathrm{Div}(X)=0$ and the only Weil divisor on $X$ is the zero divisor (Weil divisor normal noetherian scheme).

The morphism $i$ is the spectrum of the $k$-algebra homomorphism $k[t]\to k$, $t\mapsto 0$ (The underlying space of an affine spectrum, Morphisms of schemes); on global sections $i^{\mathrm{\#}}(t)=0\in k$. Its image is the origin: the unique prime $\mathfrak{p}=(0)$ of $k$ pulls back to $(i^{\mathrm{\#}}{)}^{-1}(0)=(t)$, so $i^{-1}(Z)=X$.

A pullback of all meromorphic functions is induced when $f^{\mathrm{\#}}$ carries every stalkwise nonzerodivisor section to a stalkwise nonzerodivisor. Pullback of a particular Cartier divisor requires only an admissible datum $a_i\mathrm{/}{s}_i$ whose pulled-back numerators and denominators are regular. For an effective Cartier divisor, its pullback is defined exactly when its pulled-back regular local equations remain regular (Pullback of a Cartier divisor).

A section $g\in {\mathcal{O}}_X(V)$ defines an effective Cartier divisor on $V$ only when multiplication by $g$ is injective on the local rings, i.e. when $g$ is a nonzerodivisor (Effective cartier divisor, Effective Cartier divisors are closed subschemes cut out by regular equations).