Let $k$ be a commutative ring and let $C$ be a flat module over $k$. Let $M$ be a module and let $A,B \subseteq M$ be two submodules. We get a pullback diagram:
where $s, i, j, t$ are inclusions. If we tensor by $C$ we get the diagram: However is this a pullback diagram? I cannot work out how to define the unique morphism.
Sorry about the size of the pictures.