Please answer all parts. Thank you!
20. Let R be a commutative ring with identity. We define a multiplicative subset of R to be a subset S such that 1 S and ab S if a, b E S. Define a relation ~ on R × S by (a, s) ~ (a, s') if there exists an s"e S such that s* (s,a-sa,) a. 0. Show that ~ is an equivalence relation on b. Let a/s denote the equivalence class...
Im wondering how to do b).
(6) We define the set of compactly supported sequences by qo = {(zn} : there exists some N > 0 so that Zn = 0 for all n >N). We define the set of compactly supported rational sequences by A={(za) E ao : zn E Q for all n E N). (a) Prove that A is countable (b) Prove that for 1 S p<oo the set A is dense in P. You may use...
Im wondering how to do b).
(6) We define the set of compactly supported sequences by qo = {(zn} : there exists some N > 0 so that Zn = 0 for all n >N). We define the set of compactly supported rational sequences by A={(za) E ao : zn E Q for all n E N). (a) Prove that A is countable (b) Prove that for 1 S p<oo the set A is dense in P. You may use...
1. Assume that S is an open subset of R", and that f, g: S R" are functions of class C in S. Prove that := f.g : S R is of class C, and that - D g) (Df)'g + (Dg)t (8) where T denotes "transpose."
1. Assume that S is an open subset of R", and that f, g: S R" are functions of class C in S. Prove that := f.g : S R is of class...
Given any real number x (0,1), let represent the normalized decimal expansion of x. Now define the set Prove S is a dense subset of [0,1]. We were unable to transcribe this imager = 0.2112.03... 21 +22 +13 + ... +.In S = (ce[0, 1] : lim 10
2. Assume the group G acts on the set S. For E S, define Then G is a subgroup of G , which is called the stabilizer of r. The set is called the orbit of r (a) Consider the map ф' G S, defined by фг (g) :-9-x. Prove that there is one map (and only one) : G/G, S such that Vz ยู่'z q (where q: G -G/G, is the quotient map). (b) Prove that is injective. (Hint:...
(4) Let(an}n=o be a sequence in C. Define R-i-lim suplanlì/n. Recall that R e [0,x] o0 is the radius of convergence of the power series Σ a (z 20)" Assume that R > 0 (a) Prove that if 0 < ρ < R, then the power series converges uniformly on the closed (b) Prove that the power series converges uniformly on any compact subset of the disk Ix - xo< R
(4) Let(an}n=o be a sequence in C. Define R-i-lim...
Question 5) (8 points) Consider the following subset S = {A € M3(R): AT = A, and every diagonal element of A is 0} (In words, S is the set of all symmetric 3 x 3 matrices that all have all O's on their diagonal) (a) Prove that S is a subspace of M3(R) (b) Determine a basis for S and state the dimension of S
QUESTION C.
(a) Let k be a field and let n be a positive integer. Define what is meant by a monomial ideal in k[x,...,zn]. 2. (b) State what it means for a ring R to be Noetherian. (c) State Hilbert's basis theorem. Give a proof of Hilbert's basis theorem using the fact if k is a field the polynomial ring kli,..., In] is Noetherian. 1S
(a) Let k be a field and let n be a positive integer. Define...
5. Let S be a non empty bounded subset of R. If a > 0, show that sup (as) = a sup S where as = {as : ES}. Let c = sup S, show ac = sup (aS). This is done by showing (a) ac is an upper bound of aS. (b) If y is another upper bound of as then ac S7 Both are done using definitions and the fact that c=sup S.