I'm working on prime an maximal ideals. My partner and I are studying for our final exam and got conflicting answers.
The question was to find all of the prime and maximal ideals of \mathbb Z_7. My answer was that because a finite integral domain is a field, the prime and maximal ideals...
This is my progress so far...
Well, call the matrix M. Then you want that (Mx+B)^2=Mx+B in M2R[x][/math]. So we just have to perform the calculation and see what we get,
(Mx+B)^2=MxMx + MxB + BMx+B^2 = M^2x^2+(MB + BM)x + B^2 (the x-terms act like this (commutatively) by definition)...
My professor gave us this query at the end of class, it contained two parts.
1. Show a ring is idempotent
2. Consider the degree one polynomial f(x) is an element of M2(R)[x] given by
f(x) = [0 1
______0 0]x + B
(so f(x) = the matrix []x + B).
For which B is an element of M2(R), if any...
I'm working with elementary rings, and my professor gave me about ten of these to start but it seems like a lot of work with how he managed it. I know you guys don't answer homework so I chose so I can do the others. Any help would be greatly appreciated, the groups were easy but the rings are a...
Original Query: I'm beginning to look at rings for the first time and was given this to start with:
Let R = < R, +, *> (* means multiplication) be a ring, and let I < R be an additive subgroup of < R, + >.
Consider the set of cosets
R/I = {a + I: a is an element of R}
equipped with...
Wait did I mess myself up...the original equation is % (f) = f '' - f so the homomorphism doesn't change that much, it just becomes (f + g) '' - (f + g) = f '' + g '' - f - g. Well any functions in the reals added to other functions of the reals gets you another function of the real, so % is a...
Nevermind, lol, that's what a homomorphism is.
I said the kernel is just all of the constants:
f ' (x) = 0
(Integrate)
f (x) = C
How about if we have a different function, % : G to G defined by % (f) = f '' - f ?
Homomorphism?
(f + g) '' + (f + g) = f '' + g '' + f + g. Well any...
Of course not, the constants are the only functions with no change in how they change over time, that's why they are constants.
But how do I know to look for the zero derivatives?
Let G denote the set
G = {f : R → R | f is infinitely differentiable at every point x ∈ R}.
Prove that G is a group under addition. Is G a group under multiplication? Why or why
not?
I have proved this after much trouble, using the axioms of group theory, and I think I understand the...
Show that Q/Z is isomorphic to the multiplicative group U∗ consisting of all roots of unity
in C. (That is, U∗ = {z ∈ C|zn= 1 for some n ∈ Z+}.)
I don't really understand how to prove this isomorphism
H is CG(a) in this case. The operation m' is just * (written g*h or just gh).
We want to show * : CG(a) x CG(a) -> CG(a)
This means that for any g, h in CG(a)
We need to show gh or g * h is in CG(a)
g in CG(a) means what? ga = ag (commutative)
similarly ha = ah
How do we show gh is...
Alright, so I showed this:
We have that if r+Z is an element of R/Z, and the order of r + Z = n, nr is an element of Z implies that r is an element of Q.
But what about this: Show that Q/Z is isomorphic to the multiplicative group U∗ consisting of all roots of unityin C. (That is, U∗ =...