Recent content by Rederick

Thanks for the direction, I think I got something that will work.

Homework Statement Let R = Q[x]/(x^2-4x+4). Show that R has infinitely many nilpotent elements. Homework Equations The Attempt at a Solution I see that x^2-4x+4 = (x-2)(x-2) and that a nilpotent means a^2 = 0. Other than that, I'm not sure where to start.

Homework Statement Let R = Z[x] be a polynomial ring where Z is the integers. Let I = (x) be a principal ideal of R generated by x. Prove I is a prime ideal of R but not a maximal ideal of R.Homework Equations The Attempt at a Solution I want to show that R/I is an integral domain which...

Thinking more about it,T={(100a,100b)|a,b are elements in R} is a sub-ring but not ideal since (100a,100b)(1/2,1/2)=(50a,50b) which is not in T. How would I say that in general to show that there are only 4?

Homework Statement Let S be a ring = RxR (real#,real#). Find all the ideals in RxR. Homework Equations We were told that there are only 4.The Attempt at a Solution I can only think of these 4 sub-rings of S, (R,0), (0,R), (R,R) and (0,0). And each seem to be ideal. Are these the correct 4...

Thank you. Got it

Homework Statement Let R be an integral domain with identity element 1. Show that there at most two elements "a" in R such that a^2=1 Homework Equations The Attempt at a Solution Being an integral domain implies that if ab=0, then a=0 or b=0. a^2=1 implies a=1. Then (a)(a) =...

Homework Statement If G is an abelian simple group then G is isomorphic to Zp for some prime p (do not assume G is a finite group).Homework Equations In class, we were told an example of a simple group is a cyclic group of prime order.The Attempt at a Solution Let G be an abelian simple...

Homework Statement Let (X, d1) and (Y,d2) be metric space. Define a function d:(X x Y)x(X x Y) to R by d((x1,y,1), (x2,y2))=min{d(x1,x2),d(y1,y2)}. Is d a metric on X x Y? Explain Homework Equations N/A The Attempt at a Solution Is it enough to say that the min for...

Thank you Fredrik. I got it.

Here's what I did so far... Let x=(x1,x2..xn) and y=(y1,y2..yn) in R. Assume x,y not = 0. Then (x+y)dot(x+y) = sum(x^2+2xy+y^2) >= 0. Then I rewrote it as sum(x^2 +y^2) >= sum(2xy). Using Cauchy Schwartz Inequality, sum(2xy) = 2|x dot y| <=2( ||x|| ||y||). So now I have this: sum(x^2)...

I expanded (x+y),(x+y) and got x^2+y^2 > 2xy then replaced 2xy with 2|x,y| but now I'm stuck. I need to get it to ||x+y|| <= ||x|| + ||y||. Am I close?