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- Jun 22, 2012

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I would appreciate some help with Theorem 3.2 which reads as follows: (see attachment page 7)

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Theorem 3.2. A tensor product of M and N exists.

(M and N are modules)

Proof: Consider \(\displaystyle M \times N \) simply as a set. We form the free R-module on this set:

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

(This is an enormous R-module. If \(\displaystyle R = M = N = \mathbb{R} \) then \(\displaystyle F_{\mathbb{R}} ( \mathbb{R} \times \mathbb{R}) \) is a direct sum of \(\displaystyle \mathbb{R}^2 \)-many copies of \(\displaystyle \mathbb{R} \). The direct sum runs over all pairs of vectors, not just pairs coming from a basis, and for modules bases do not even usually exist)

Let D be the submodule of \(\displaystyle F_R (M \times N) \) spanned by all the elements:

\(\displaystyle \delta_{(m + m', n)} - \delta_{(m,n)} - \delta_{(m',n)} , \ \ \delta_{(m, n + n')} - \delta_{(m,n)} - \delta_{(m,n')} , \ \ \)

\(\displaystyle \delta_{(rm,n)} - \delta_{(m, rn)} , \ \ r \delta_{(m,n)} - \delta_{(rm,n)} , \ \ r\delta_{(m,n)} - \delta_{(m,rn)} , \ \ \)

... ...

My problem with this is that Conrad does not define the symbols \(\displaystyle \delta_{(m,n)} \) - can someone help me with their meaning, including the meaning of them in the direct sum above?

Such a clarification would be most helpful.

Peter