Let $R$ be a commutative ring and $M$ be an $R$-module.
Let $T(M)$ be the tensor algebra of $M$.
Then, what is $S(M)$ (symmetric algebra and $S^k(M)$?
Some articles define $S(M)$ as a quotient of the tensor algebra while others define $S(M)$ as a subalgebra of the tensor algebra. Wikiepdia article specifices the difference between two, but I completely don't understand that article since I'm completely not familar with symmetric algebra. What's the intrinsic difference between two? (Q1)
Moreover, I am really uncomfotable with the definition of symmetric algebra as a quotient of the tensor algebra when dealing with $k^{th}$-symmetric power.
To be specific, my text(Dummi&Foote) defines it via graded ring. So that he defines $\{S^k(M)\}$ as a sequence of abelian groups whose direct sum of $T(M)/I$ where $I$ is an ideal generated by $v\otimes w - w\otimes v$ and other properties that asserts $T(M)$ is a graded ring. I'm uncomfortable with this definition since $S^k$ cannot be well-defined. There maybe two sequences $\{A_n\}$ and $\{B_n\}$ satisfying the hypothesis of the definition of graded ring. How do I exactly and dry-formally define $S^k$? (Q2)
Great thanks in advance. Please help me!