Let R be an integral domain with elements a,b in R and <a>,<b> the corresponding principal ideals. Prove that <a>=<b> if, and only if, a=bu for some unit u in U(R).
proof
if a=bu, then ar=bur. Since ur in R, call it s. So ar=bs for some s in R. Therefore a times some elements in r is equal...
Let R be a ring with ideals I, J, and P. Prove that if P is a prime ideal and I intersect J is a subset of P, then I is a subset of P or J is a subset of P.
1. Homework Statement
True or false? If f and g are continuous at 0 and f(1/(2n+7))=g(1/(7-2n)) for all positive integers n, then f(0)=g(0).
2. Homework Equations
lim x->0 f(x)=f(0)
lim x->0 g(x)=g(0)
3. The Attempt at a Solution
NO CLUE. My intuition says false.
Homework Statement
True or false? If f and g are continuous at 0 and f(1/(2n+7))=g(1/(7-2n)) for all positive integers n, then f(0)=g(0).
Homework Equations
lim x->0 f(x)=f(0)
lim x->0 g(x)=g(0)
The Attempt at a Solution
NO CLUE. My intuition says false.
Homework Statement
Suppose that f(x)>=0 in some deleted neighborhood of c, and that lim x->c f(x)=L. Prove that lim x->c sqrt{f(x)}=sqrt{L} under the assumption that L>0.
Homework Equations
When 0<|x-c|<delta, |f(x)-L|<epsilon.
The Attempt at a Solution
When, 0<|x-c|<delta...
Right the homomorphism part is easy now. Am I able to use the pigeonhole principle for the isomorphic part? That is, are HxN and HN the same size? It seems like they are since H intersect N is only the identity.
I'm lost on this one. It doesn't make sense how the number of left cosets corresponds to the normality. #gH=#Hg doesn't seem like it necessarily means that gH=Hg.
Let G be a group, H a normal subgroup, N a normal subgroup, and H intersect N = {e}. Let H x N be the direct product of H and N. Prove that f: HxN->G given by f((h,n))=hn is an isomorphism from HxN to the subgroup HN of G.
Hint: For all h in H and n in N, hn=nh.