@JonBeardsley I cut my teeth on Herstein's Abstract Algebra plus Atiyah-MacDonald for the commutative algebra part (Herstein's is not very good there in my opinion). The exercises are tough but in my opinion this is a plus for self-study. In Pisa it was the standard basic algebra book (together with Di Martino, but that's only in Italian)

Also, the old Italian edition of Herstein's book has the most surrealist cover I've ever seen in a math book... amazon.it/Algebra-I-N-Herstein/dp/8835954797/…

@JonBeardsley i didn't use it the first time i learned algebra, but when i went back to relearn things by myself i found this book by aluffi that i really enjoyed (partially because he introduces universal properties and simple category-theoretic ideas early on, so highly dependent on the student...)

I agree, that's a cool book

Let $R$ be a commutative ring spectrum, $A$ be a commutative $R$-algebra, $E$ be an $R$-module. Suppose $A$ is $E$-local, i.e. $[W,A]=0$ for any $R$-module $W$ such that $E\wedge W\simeq *$. Is $THH^R(A)$ $E$-local?

I second Aluffi for general abstract algebra, and Atiyah-MacDonald for commutative algebra.

@BrunoStonek @BrunoStonek I don' t think so unless the $E$-localization functor is smashing. For instance, if $E_1$ is Morava E-theory of height 1 (i.e. p-completed K-theory), $ THH(E_1)\simeq E_1\wedge \Sigma(E_1)_{\mathbb{Q}}$ is not $K(1)$-local.

@JonBeardsley i also used herstein's smaller algebra book (not 'topics in algebra'). it's a great book: it's full of exercises, and the book is arranged so that all the exercises are trivialized by the next section, so the semester is lots of hard work regularly punctuated by 'but look! there's a better way!'

which mb sounds frustrating, but it worked for me

@GeoffroyHorel I think you meant $\vee$, not $\wedge$. At any rate, yeah, I'm ready to assume $E$ is smashing

@BrunoStonek Yes I meant \vee. If E is smashing then E-local objects are stable under tensor product and homotopy colimits. Using the cyclic bar construction, you immediatly see that THH(A) is E-local

@JonBeardsley Maybe Isaacs book on algebra is a good option. It's not very categorial, but it's the best book on general algebra I've ever seen.

Does anyone know the first time that the gamma filtration for algebraic K-theory was introduced? I keep hearing that it was introduced by Grothendieck , but I couldn't find the corresponding paper.

SGA 6