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I am trying to understand an exposition on Tensor products by Keith Conrad. In the proof of Theorem 3.2 on page 7 it considers the free $R$-module on the set $M \times N$:

$$F_R(M \times N) = \bigoplus_{(m,n) \in M \times N} R\delta_{(m,n)}$$

What exactly does this notation mean? I'm guessing that $R\delta_{(m,n)}$ is suppose to be shorthand for the module with a single generator, meaning that $R\delta_{(m,n)} = \{r(m,n)| r \in R\}$ where $(m,n)$ is taken as a formal symbol. But then why I don't understand why the notation $\delta_{(m,n)}$ is used over $(m,n)$.

user782220
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1 Answers1

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Yes, you could also write $(m,n)$ instead of $\delta_{(m,n)}$. It is just a matter of taste. Perhaps Keith made this distinction so that the laws $m \otimes n + m' \otimes n = (m+m') \otimes n$ in $M \otimes_R N$ do not get confused with $(m,n) + (m',n) = (m+m',n)$ in $M \times N$ (which is wrong).

Yes, $R \delta$ denotes the free $R$-module whose generator is called $\delta$.

PS: See here for an alternative construction of $M \otimes_R N$ which doesn't need free modules at all.

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Martin Brandenburg
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