let R be a ring; X a non-empty set and (F(X, R), +, *) the ring...

let R be a ring; X a non-empty set and (F(X, R), +, *) the ring of the functions from X to R. Show directly the associativity of the multiplication of F(X, R). Assume that R is unital and commutative. show that F(X, R) is also unital and commutative.

Expert Solution

Colby Messinger answered 2 years ago

Let X be a non-empty set and R⊆X × X be an equivalence relation. Prove that...

Let X be a non-empty set and R⊆X × X be an equivalence relation. Prove that X / R is a partition of X.

2 Let F be a field and let R = F[x, y] be the ring of...

2 Let F be a field and let R = F[x, y] be the ring of polynomials in two variables with coefficients in F. (a) Prove that ev(0,0) : F[x, y] → F p(x, y) → p(0, 0) is a surjective ring homomorphism. (b) Prove that ker ev(0,0) is equal to the ideal (x, y) = {xr(x, y) + ys(x, y) | r,s ∈ F[x, y]} (c) Use the first isomorphism theorem to prove that (x, y) ⊆ F[x, y]...

Let R be a commutative ring with unity. Prove that f(x) is R[x] is a unit...

Let R be a commutative ring with unity. Prove that f(x) is R[x] is a unit in R[x] iff f(x)=a is of degree 0 and is a unit in R.

Let R be a ring (not necessarily commutative), and let X denote the set of two-sided...

Let R be a ring (not necessarily commutative), and let X denote the set of two-sided ideals of R. (i) Show that X is a poset with respect to to set-theoretic inclusion, ⊂. (ii) Show that with respect to the operations I ∩ J and I + J (candidates for meet and join; remember that I+J consists of the set of sums, {i + j} where i ∈ I and j ∈ J) respectively, X is a lattice. (iii) Give...

Let X be a non-empty set and P(X) its power set. Then (P(x), symetric difference, intersection)...

Let X be a non-empty set and P(X) its power set. Then (P(x), symetric difference, intersection) is a ring. Find a non-trivial ideal of P(X).

c) Let R be any ring and let ??(?) be the set of all n by...

c) Let R be any ring and let ??(?) be the set of all n by n matrices. Show that ??(?) is a ring with identity under standard rules for adding and multiplying matrices. Under what conditions is ??(?) commutative?

Let A and B be two non empty bounded subsets of R: 1) Let A +B...

Let A and B be two non empty bounded subsets of R: 1) Let A +B = { x+y/ x ∈ A and y ∈ B} show that sup(A+B)= sup A + sup B 2) For c ≥ 0, let cA= { cx /x ∈ A} show that sup cA = c sup A hint:( show c supA is a U.B for cA and show if l < csupA then l is not U.B)

Let x be a set and let R be a relation on x such x is...

Let x be a set and let R be a relation on x such x is simultaneously reflexive, symmetric, and antisymmetric. Prove equivalence relation.

6. (a) let f : R → R be a function defined by f(x) = x...

6. (a) let f : R → R be a function defined by f(x) = x + 4 if x ≤ 1 ax + b if 1 < x ≤ 3 3x x 8 if x > 3 Find the values of a and b that makes f(x) continuous on R. [10 marks] (b) Find the derivative of f(x) = tann 1 1 ∞x 1 + x . [15 marks] (c) Find f 0 (x) using logarithmic differentiation, where f(x)...

. Let U be a non-empty set. For A and B subsets of U, define the...

. Let U be a non-empty set. For A and B subsets of U, define the relation A R B if an only if A is a proper subest of B. a. Is R reflexive? Prove or explain why not. b. Is R symmetric? Prove or explain why not c. Is R transitive? Prove or explain why not. d. Is R antisymmetric? Prove or explain why not. e. Is R an equivalence relation? Prove or explain why no

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