Milnor square

Remarks

Example: let $R \subseteq S$ be rings and let $I$ be an ideal of $S$ contained in $R$, hence also an ideal of $R$. Then the square with $R \to S$, $R \to R/I$, $S \to S/I$ and $R/I \to S/I$ is a Milnor square: $S \to S/I$ is surjective, and $(s, r + I)$ with $s + I = r + I$ forces $s \in R$, so $R = S \times_{S/I} R/I$. Example: for a field $k$, the ring $k[x, y]/(xy)$ is the pullback of $k[x] \to k \leftarrow k[y]$, both maps sending the variable to $0$. Non-example: the square with $R = R_1 = R_2 = \Z$ and $R' = \Z/2\Z$, all maps natural, is commutative with surjective $g_i$, but it fails clause 1, since the pullback $\set{(a, b) \in \Z^2 : a \equiv b \pmod 2}$ is not $\Z$.