Homework Statement
Suppose that R and S are two rings, M, is a (R-S) bi-module and N is a left R-module. Show that M \otimes N has the structure of a left S-module.
The Attempt at a Solution
Well, M\otimes N is an Abelian group, so it's enough that I define a scalar product on...
No, I know the definition, the definition is definition, I have nothing to say about that. Here's my question: Suppose I have given to you two elements of a ring. like b and a. How do you find if b|a or not? you must find an element like r that rb=a, but how do you find that particular r? If we...
Thanks for the answer, I knew what b|a meant but I wondered how we could find b|a if there's no division algorithm. I mean for polynomials and integers and other rings that have a division algorithm it's easy, but for other rings it must be hard to show that a=br. Am I right or we can always...
Thanks for the answer.
Is every P.I.D ring a Euclidean domain? Because if it's not a Euclidean domain and we don't have the division algorithm, how can we find if b|a in the ring?
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Anyway, I used micromass's answer to...
I know that Z/pZ is a field therefore pZ must be a maximal ideal because of the theorem that states "R/I is a field if and only if I is a maximal ideal" but I want to see a direct proof of it because I hope it would give me an idea how to prove something is a maximal ideal in a general field...
Actually proving that every infinite cyclic group has such a group is a bit hard, I could prove it in an easier way.
If G is cyclic, then all elements of G can be generated by a specific element like a. so we have G=<a> iff g = an for any g in G. Now, every subgroup of G is cyclic too, this can...
Actually I'm stupid today, it happens once in a while that I get extremely lazy and stupid in mathematics, but today I came up with a bizarre thing in abstract algebra that I couldn't find my mistake on my own and I'm not sure whether what I've concluded is true or wrong, I was proving another...
Well, We create the maps in a way that the diagram commutes, that's how I obtain α
,β and γ. You're right though, because I haven't precisely shown that γψ=π (It's really very obvious that the rest of the diagram commute) but it's not immediately followed from γ
=πψ-1 that we have γψ=π, so I'll...
I've attached the problem statement and the solution in a pdf file, please check it and see if I've done it correctly. I'm new to abstract reasoning, I've only had a one semester introduction to group-theory and parts of ring theory based on baby Herstein, so I need others to check my proofs to...
Hey, today I read about the concept of an exact functor. Can we say that, according to this theorem, the tensor product is an exact functor? Is this what this theorem wants to tell us?
Well, I'm convinced that defining 0^0 is ambiguous, but I'm not convinced why 0!=1 is well-defined yet. I read the page you said, I have no problem with it, it says that in general they define prod({})=1. It's a very reasonable definition, but my question is, is it the ONLY way that we can...
well, I read the page that ILS had given, I'm still not convinced why 0!=1 is well-defined (we define it this way!) but 0^0=1 is ill-defined.
I found this interesting explanation on wikipedia:
http://en.wikipedia.org/wiki/Exponentiation#Zero_to_the_zero_power
It seems that we should blame...