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Thanks, micromass! I think I've got it. If you wouldn't mind, please let me know if this looks alright. Define h: M \times K \rightarrow M by h(m,k)=mk. This map is bilinear, so it factors through the tensor map M \times K \rightarrow M \otimes K. Thus there is a unique (irrelevant?)...

Let M be a module over the commutative ring K with unit 1. I want to prove that M \cong M \otimes K. Define \phi:M \rightarrow M \otimes K by \phi(m)=m \otimes 1. This is a morphism because the tensor product is K-linear in the first slot. It is also easy to show that the map is surjective...

Very nice quotes, Adyssa! I'm going to put the first up on my wall. Thanks for the replies everyone!

Are there examples of modern, influential mathematicians who did not show unusual mathematical talent before college? Because today it seems like the field is pretty much closed off to college freshmen first encountering abstract mathematics, no matter how much fascination and interest they...

Thank you very much! I guess I was thinking incorrectly that the order of x was also the order of the quotient group. Back to the drawing board!

Hello! If anybody has a minute, I'd appreciate a quick look-through of my proof that a finite abelian group can be decomposed into a direct product of cyclic subgroups. I'm new to formal writing (as well as Latex) and all feedback is greatly appreciated! Thanks in advance for your time...

Thanks, Ben. To clarify, do you mean that we set ##V=R^n ## so that the group of linear maps on ##V## is the set of ##n##x##n## matrices? Also, do you know of a textbook that explains exterior algebra (from the module perspective) and its connections to the determinant from the ground up?

Is there a nice way to show that Det(AB)=Det(A)Det(B) where A and B are n x n matrices over a commutative ring? I'm hoping there is some analogue to the construction for vector spaces that defines the determinant in a natural way using alternating multilinear mappings... Otherwise would...

Why Denote Group Operation with Multiplication?? When groups are introduced in most abstract algebra texts, the operation is denoted by multiplication or juxtaposition and addition notation is reserved for abelian groups. This seems to cause a lot of unnecessary confusion. Professors often...

I'm a prospective math major going to college next fall. Are there any cheap or free math programs available for a graduating high school senior during the summer? Research sounds exciting but I'd be fine with just learning material. By the end of the year I'll have learned basic abstract...

Read Basic Mathematics by Serge Lang

Spivak's Calculus is not quite at the level of introductory real analysis, so moving to a second course in real analysis would be quite a jump. If I were you, I would first learn single variable analysis. Rudin's Principles of Mathematical Analysis essentially boils Spivak's Calculus down to...

Is Lang a good text to read through and learn the material? Most of the reviews on Amazon imply that it is great as a reference but not useful as a textbook. I've gotten an overview of algebra from Herstein's Topics in Algebra, though I am shaky on the material in the later chapters. I've...

I've heard that Lang is full of typos and is a very boring read. Though if boring just means elegant and terse I'd be happy to pick it up, as I enjoyed Rudin. Does Lang integrate category theory and universal algebra throughout the text? espen180, unfortunately the closest library that has...

I'm looking for a textbook that covers all of the standard undergrad algebra topics but from a more modern perspective. For example, most books reprove the isomorphism theorems for groups, rings, modules, instead of showing that all of these structures are universal algebras. Ideally the text...