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- Jun 22, 2012

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I am reading and trying to fully understand Keith Conrad's paper: Tensor Products I. These notes are available at Expository papers by K. Conrad or the specific paper at http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/tensorprod.pdf.

Conrad's Theorem 3.3 (see attachment - page 10) is important since it, to an extent at least, begins to define the nature of elementary tensors \(\displaystyle m \oplus n \) in the tensor product \(\displaystyle M \oplus_R N \) where M and N are R-modules.

Theorem 3.3 and its proof read as follows:

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The tensor product \(\displaystyle M \oplus_R N \) is spanned linearly by the elementary tensors \(\displaystyle x_i \oplus y_j \)

Write \(\displaystyle m = \sum_i a_ix_i \) and \(\displaystyle n = \sum_j b_jy_j \), where the \(\displaystyle a_i s \) and \(\displaystyle b_j s \) are 0 for all but finitely many i and j.

From the bilinearity of \(\displaystyle \oplus \) we have:

\(\displaystyle m \oplus n = \sum_i a_ix_i \oplus \sum_j b_jy_j = \sum_{i,j} a_ib_jx_i \oplus y_j \)

is a linear combination of the tensors \(\displaystyle x_i \oplus y_j \).

Since every elementary tensor is a sum of elementary tensors, the \(\displaystyle x_i \oplus y_j 's \) span \(\displaystyle M \oplus_R N \) as an R-module.

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In the above Conrad assumes that the R-modules M and N have spanning sets \(\displaystyle \{ x_i \}_{i \in I} \) and \(\displaystyle \{ y_j \}_{j \in J} \).

Do all modules have spanning sets? (they do not necessarily have bases - which is a similar concept - unless they are free modules, of course.)

Is this simply a case of taking more and more elements in the spanning set ... even up to all the elements of the module? But then how do we guarantee that for elements m and n from M and N that we can write:

\(\displaystyle m = \sum_i a_ix_i \) and \(\displaystyle n = \sum_j b_jy_j \), where the \(\displaystyle a_i s \) and \(\displaystyle b_j s \)

Can someone please clarify this issue for me. I would appreciate guidance on this matter.

Peter

Conrad's Theorem 3.3 (see attachment - page 10) is important since it, to an extent at least, begins to define the nature of elementary tensors \(\displaystyle m \oplus n \) in the tensor product \(\displaystyle M \oplus_R N \) where M and N are R-modules.

Theorem 3.3 and its proof read as follows:

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**Theorem 3.3.**Let M and N be R-modules with respective spanning sets with respective spanning sets \(\displaystyle \{ x_i \}_{i \in I} \) and \(\displaystyle \{ y_j \}_{j \in J} \).The tensor product \(\displaystyle M \oplus_R N \) is spanned linearly by the elementary tensors \(\displaystyle x_i \oplus y_j \)

*Proof:*An elementary tensor in \(\displaystyle M \oplus_R N \) has the form \(\displaystyle m \oplus n \).Write \(\displaystyle m = \sum_i a_ix_i \) and \(\displaystyle n = \sum_j b_jy_j \), where the \(\displaystyle a_i s \) and \(\displaystyle b_j s \) are 0 for all but finitely many i and j.

From the bilinearity of \(\displaystyle \oplus \) we have:

\(\displaystyle m \oplus n = \sum_i a_ix_i \oplus \sum_j b_jy_j = \sum_{i,j} a_ib_jx_i \oplus y_j \)

is a linear combination of the tensors \(\displaystyle x_i \oplus y_j \).

Since every elementary tensor is a sum of elementary tensors, the \(\displaystyle x_i \oplus y_j 's \) span \(\displaystyle M \oplus_R N \) as an R-module.

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**Now my issue/problem ...**In the above Conrad assumes that the R-modules M and N have spanning sets \(\displaystyle \{ x_i \}_{i \in I} \) and \(\displaystyle \{ y_j \}_{j \in J} \).

Do all modules have spanning sets? (they do not necessarily have bases - which is a similar concept - unless they are free modules, of course.)

Is this simply a case of taking more and more elements in the spanning set ... even up to all the elements of the module? But then how do we guarantee that for elements m and n from M and N that we can write:

\(\displaystyle m = \sum_i a_ix_i \) and \(\displaystyle n = \sum_j b_jy_j \), where the \(\displaystyle a_i s \) and \(\displaystyle b_j s \)

**are 0 for all but**__finitely many__i and j.Can someone please clarify this issue for me. I would appreciate guidance on this matter.

Peter

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