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Over the past week, I read this secret blogging seminar post concerning the diamond lemma, which got me to reading about Bergman's paper on the diamond lemma.

Now suppose you have a free $A$-module $M$ over some commutative ring $A$, with basis $\{e_i\}_{i\in I}$. Standard methods using the tensor product yield the fact that the basis of the symmetric algebra $S(M)$ is given by monomials $e_{i_1}\dots e_{i_k}$ for $i_1\leq\cdots\leq i_k$ for some linear ordering on $I$, and similarly for the exterior algebra $\Lambda(M)$.

How can the same results be reached using Bergman's diamond lemma, and avoiding the tensor product arguments? I'm hoping to see how it can be applied in these situations. How can I see what the possible ambiguities are to check? Thank you.

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