So without giving excessive background, I was working on baby Spivak & got to the stuff about exterior algebras, & now I'm going through the section on tensor algebras in my copy of Dummit/Foote to learn more. They define it (as usual?) as the R-algebra $\bigwedge(M) = T(M)/A(M)$ obtained by taking the quotient of the tensor algebra T(M) by the ideal A(M) generated by all elements $m\in M$ with the property that $m\otimes m = 0$. So that got me thinking about radicals, since Hungerford described them as a way to "mod out" unwanted elements of a ring. This is how he described the unwanted radical property P
i) the homomorphic image of a P-ring (ring with property P) is itself a P-ring
ii) every ring R contains a P-ideal P(R) that contains all other P-ideals
iii) P(R/P(R)) = 0
iv) P(P(R)) = P(R)
(& in this case P(R) is A(M) and T(M) is R of course)
So you can probably guess what I'm thinking. I can believe that i) & ii) hold at least when M is a vector space, and I can show iv), it's just iii) that I'm having trouble with, although that one doesn't seem hard to believe either. Also, if A(M) is a radical of T(M), when is T(M) itself radical and when is it semisimple? Are there any books or articles out there where someone has done it like this? Is there any advantage to doing it this way?