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Let $F$ be a free module over a field k indexed by the set $(V,W)$ with $V,W\in \operatorname{Vect(k)}$. Let $H$ be the subgroup of $F$ that is generated by the following set of elements of $F$ $$ \{(v_1,w)+(v_2,w)-(v_1+v_2,w), (v,w_1)+(v,w_2)-(v,w_1+w_2), (cv,w)-(v,cw))\} $$ Then the tensor product $V\otimes W$ is defined as $F/H$ and a simple tensor, $v\otimes w$, is just a coset of $F/H$.

The above is the definition of a tensor that I'm working with. However, what I'm reading doesn't specify what a non-simple tensor is. My question is, are the non-simple tensors just the representatives of the cosets of $F/H$?

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    what do you understand by ..."indexed by the set $(V,W)$"...? – 2017-01-19
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    @janmarqz the set $(V,W)$ is every element of $V\times W$ while ignoring any of the structure. – 2017-01-19
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    If I'm understanding your question, the "non-simple" tensors are finite linear combinations $\sum r_i v_i \otimes w_i$. – 2017-01-19
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    @user4894 So any combination of simple tensors is what I'm calling (for lack of a better word) just a non-simple tensor? – 2017-01-19
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    @Bob Yes, the simple tensors are a basis for the tensor product. – 2017-01-19

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