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The pullback is a subset of the cartesian product in the category of commutative rings with unit.

What is the pullback in the category of commutative $k$-algebras? Is it the same set as in rings?

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    The forgetful functor to Set is representable, so it preserves limits (including pullback). Hence the pullback should be the set-theoretic pullback with the natural commutative k-algebra structure.2010-12-09
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    @Qiaochu I think this is most natural way to see this.2010-12-09

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