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Let h\colon A\to A' be a ring homomorphism between A,A' which are commutative rings with $1$.

Let $P,Q$ be $A$-modules. Then, are there any gaps in my following argument?

A'\otimes_{A}(P\otimes_{A}Q)=(A'\otimes_{A}P)\otimes_{A}Q=((A'\otimes_{A}P)\otimes_{A'}A')\otimes_{A}Q=(A'\otimes_{A}P)\otimes_{A'}(A'\otimes_{A}Q).

I'm concerned about this because of the proving the ring homomorphism between commutative ring with 1 induces the ring homomorphism between $K_0$ of rings.

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h:A \to A' makes A' (bi) modules into $A$ (bi) modules ("restriction"), so A' \otimes_{A} (P \otimes _A Q) is fine (it's "induction"). The associativity of $\otimes$ that you may be worried about is covered in Cartan and Eilenberg.

What you've written looks ok to me, but it is very special to the case when the rings are commutative. You're implicitly using that a left (say) $A$ module $M$ becomes an $A$ bimodule when $a \cdot m \cdot b$ is defined to be $abm$, which of course needs commutativity.

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    Well, $K_0(A)$ is usually not a ring, except when $A$ is commutative, so this is necessarily special :)2011-04-01