False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16

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False: every element of $M \otimes_{R} N$ is an elementary tensor

Statement

False claim: every element of a tensor product is an elementary tensor.

For any field $F$ , if $e_{1}, e_{2}$ and $f_{1}, f_{2}$ are the standard bases of two copies of $F^{2}$ , then

$e_{1} \otimes f_{1} + e_{2} \otimes f_{2}$

is not an elementary tensor.

Facts & Assumptions

Given: A field $F$ and two copies of $F^{2}$ .

[L1]

The two standard coordinate vectors form a basis of $F^{2}$ (The standard list $e : n \to F^{n}$ with $e_{i} (i) = 1_{F}$ and $e_{i} (j) = 0_{F}$ for $j \neq i$ is an ordered basis of $F^{n}$ ; hence $\dim_{F} F^{n} = n$ , and $F^{0}$ is the zero space with basis $\emptyset$ and dimension $0$ ).

[L2]

The four tensors $e_{i} \otimes f_{j}$ form a basis of $F^{2} \otimes_{F} F^{2}$ (The elementary tensors of two bases form the product basis of the tensor product).

Refutation

technique · direct

1.1

Suppose $e_{1} \otimes f_{1} + e_{2} \otimes f_{2}$ were elementary. By [L1], write its factors as $a e_{1} + b e_{2}$ and $c f_{1} + d f_{2}$ .

givenL1assume-hyp

2.1

Expanding the elementary tensor gives coefficients $a c, a d, b c, b d$ on the ordered basis $e_{1} \otimes f_{1}, e_{1} \otimes f_{2}, e_{2} \otimes f_{1}, e_{2} \otimes f_{2}$ . Uniqueness of coefficients in [L2] therefore yields $a c = b d = 1$ and $a d = b c = 0$ .

step 1.1L2algebra

3.1

From $a c = 1$ , both $a$ and $c$ are nonzero. Then $a d = 0$ forces $d = 0$ , contradicting $b d = 1$ . Hence the displayed tensor is not elementary and the claim is false.

step 2.1algebra∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 64 results over 20 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Haynes Miller, Lectures on Algebraic Topology I, §20 (standard reference, not scraped)

False: every element of M⊗RNM⊗R​N is an elementary tensor