Let v be a euclidean valuation on Integral Domain R.
If x , y are nonzero associates in R. Then x=yz for some unit zR.
Since v is a euclidean valuation on R, therefore by property (i)
Since z is a unit in R. Therefore it has a multiplicative inverse z-1 R.
Therefore y=xz-1
So again by property (1) of euclidean valuation,-
From Inequalities (1) and (2)
v(x)=v(y) for any nonzero associates x,y in R.
First: As I mentioned in my e-mail, a Euclidean valuation on an integral domain R is...
(9) Let E R" and let A E L(R"). Define a map f : R" -> R" by f (x) A,)v. Here (is the Euclidean inner product (a) Prove that f is a C1 map and find f'(x) (b) Prove that there exist two that f U V is a bijection on R" neighborhoods of the origin in R", U and V, such (9) Let E R" and let A E L(R"). Define a map f : R" -> R"...
8, Prove that if R is an integral domain and a E R such that a2 + 2a + 1-0 then a =-1. Then give an example of a ring that is not an integral domain for which a2a 1-0 but a f -1. 8, Prove that if R is an integral domain and a E R such that a2 + 2a + 1-0 then a =-1. Then give an example of a ring that is not an integral domain...
(4) Let R be an integral domain and o n,b,ce R such that ab = ne. Prove that b
Please help! Thank you so much!!! 2. (10 points) Let R be an integral domain and M a free R-module. Prove that if rm 0 or m 0 where r E R and m E M, then either r 0. 2. (10 points) Let R be an integral domain and M a free R-module. Prove that if rm 0 or m 0 where r E R and m E M, then either r 0.
Consider R with the usual Euclidean topology and let I = [0, 1] be the closed unit interval of R with the subspace topology. Define an equivalence relation on R by r ~y if x, y E I and [x] = {x} if x € R – I, where [æ] denotes the equivalence class of x. Let R/I denote the quotient space of equivalence classes, with the quotient topology. Is R/I Hausdorff? Is so, prove so from the definition of...
(a) Show that for > 0, the first integral equals: 0 n-0 (b) Show that 1) n=0 defines a continuous function on Ω :-R\{0,-1,-2,-3, . . . } [Hint: consider the domain R\ U000 (-n-e,-n + ε) for any small ε > 0.] (a) Show that for > 0, the first integral equals: 0 n-0 (b) Show that 1) n=0 defines a continuous function on Ω :-R\{0,-1,-2,-3, . . . } [Hint: consider the domain R\ U000 (-n-e,-n + ε)...
Problem 4. Let n E N. We consider the vector space R” (a) Prove that for all X, Y CR”, if X IY then Span(X) 1 Span(Y). (b) Let X and Y be linearly independent subsets of R”. Prove that if X IY, then X UY is linearly independent. (C) Prove that every maximally pairwise orthogonal set of vectors in R” has n + 1 elements. Definition: Let V be a vector space and let U and W be subspaces...
please answer all the questions. question 1 to question 5 Given an integral domain R we define the relatic n~on Rx (R (0]) by (a, b)~(c, d) means ad bc. We also define the following operations on R x (R\o) (a, b) + (c, d) (ad + be, bd) and (a, b) (c,d) (ac, bd). 1. Prove that ~ is an equivalence relation. 2. Prove that ~is compatible with +and . (Therefore, ~is a congru- 3. Conclude that the following...
Real analysis 10 11 12 13 please (r 2 4.1 Limit of Function 129 se f: E → R, p is a limit point of E, and limf(x)-L. Prove that lim)ILI. h If, in addition, )o for all x E E, prove that lim b. Prove that lim (f(x))"-L" for each n E N. ethe limit theorems, examples, and previous exercises to find each of the following limits. State which theo- rems, examples, or exercises are used in each case....
Q9 6. Define Euclidean domain. 7. Let FCK be fields. Let a € K be a root of an irreducible polynomial pa) EFE. Define the near 8. Let p() be an irreducible polynomial with coefficients in the field F. Describe how to construct a field K containing a root of p(x) and what that root is. 9. State the Fundamental Theorem of Algebra. 10. Let G be a group and HCG. State what is required in order that H be...