Recent content by Bleys

Ah, I forgot: the direct product includes all combinations of elements of the summands! I also kept thinking the diagonal subset was some kind of pathological example (with B=A), but of course this works for general sets. Thank you for explaining DonAntonio! :D

Thanks for your replies! I'm sorry I'm having a little trouble understanding. Isn't the cartesian product defined as the set of elements of the form (g,h). Then any subset is a set of this form as well, so it is another direct product? If it is, why aren't the summands subsets of their...

I was reading on wikipedia on direct product of groups because I wanted find out if every subgroup of G \times H is realized as a direct product of subgroups of G and H. Apparently it is not, because the diagonal subgroup in G \times G disproves this. I'm a little confused, because I thought...

I guess this would be an elementary number theory question, but it's in Advanced Algebra by Rotman, so I figured it would go here. I apologize if it's wrong. If p is an odd prime and a_{1},...,a_{p-1} is a permutation of 1,2,...,p-1 then there exist i \neq j with ia_{i} \equiv ja_{j} modp ...

ah of course, I didn't think of the fact End_{R}(S) is a subspace of End_{C}(S). I'm actually going through Schur's Lemma's proof to show End_R(S) is isomorphic to C but this was the detail I wasn't understanding. Thank you, morphism!

yes; I have this result "For a finite dimensional C-algebra R, there are only finitely many isomorphism classes of simple R-modules and they are finite dimensional"

Let R be a finite dimensional C-algebra (C=Complex numbers) and S a simple R-module. Why does it follow that End_{R}(S) is also finite dimensional (as C-vector spaces, I'm guessing)? I'm not really sure how to construct a basis for it using one of S, and there's probably another reason for it...

?? \phi takes a point (s_{0}:s_{1}:s_{2}) to (s_{0}:s_{1}) and/or (s_{2}:s_{0})

So in class today the lecturer gave a regular map on the set V(s_{1}s_{2}-s_{0}^2) in projective 2-space to projective 1-space by \phi = (s_{0}:s_{1})=(s_{2}:s_{0}). I'm confused. Is that another representation of the "function"? (Meaning they map to the same point classes?) or is it an...

Hello there. I'm currently applying to several Master's programs in mathematics around the UK (and also elsewhere in Europe). I'm interested in algebra or (algebraic) number theory, but I'm keeping my options open to explore more areas before I consider, and apply for, doctorate studies (which...

thank you, Mark!

Hello there; I have a very short question about client class access: Suppose in a supplier class an instance variable is declared private (and is used in the constructor), and a void method in the same class (declared public) changes this variable. In the client class, is calling this method...

Thank you both for the reply! Ok so the definition I'm interested in is (2). But why the need to call it affine n-space? Is there background or reason to this? I know "affine" means 'analogous' or 'similar' and I could understand from definition (1) why it would be called that (the analogy of...

I need some help about the definition of this. I cannot find it properly anywhere. This concerns algebraic geometry. What is an affine n-space? In some books, k is a field and it says this is just the point space k^n. But on wikipedia there is a whole abstract definition of it, and I don't...

aah, thanks a bunch! I had thought about "picking" certain elements of the subcover such that y was in V_{i}, but then I was wondering if the remaining U_{i} would still cover X, so I was getting confused (forgetting slightly that I was dealing with Cartesian products!). Thanks for your...