7. a) Let R be an integral domain and a € R. Show that the function f : R[x] → R given by f(p(x)) = p(a) is a ring homomorphism. This is often called the evaluation homomorphism at a.. b) Find Ker f. It has a nice description in words.

Q: Let A be an associative algebra. (a) Show that for any a E A the map da : A A defined by da(b) = ab…

A: Let A be an associated algebra.  To prove that for any a∈A, the map da:A→A defined by dab=ab-ba is a…

Q: 4. Let R be a ring. (a) Let x, y, z € R. Prove that, if x | y and y | z, then x | z. (b) Prove that…

A: Let R be  a  ring .4) a)  Let  Suppose that   divides y and  y  divides z.We  have  to show that x…

Q: -se A, B are submodules (that is, they are isom : M/A→M/B and show pism, then use the FHT

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Q: Which of the following maps C[X] → C[X] are ring homomorphisms? Select all that apply: O The map…

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Q: 2. Let s-{(소)아 2a S = 26 2a Assume that S is a subring of M2(Q). Prove that S is isomorphic to Q[V2]…

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Q: Problem 1. Let R be a ring. Construct an isomorphism between¹ R[X,Y]/(XY) and the subring A CR[X] ×…

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Q: Exercise 7.24. (a) Suppose : Z36 → G is a surjective homomorphism, where |G| (b) (a) Suppose : Z25 →…

A: (a) Here we have given that ϕ:ℤ36→G is a surjective homomorphism, where G=12. Now, by the first…

Q: M n of ring (5 Marks) (b) Let (R,+,) and (R', +', '') be any two rings, let f: RR' be a ring…

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Q: 2. If f: R→ S is a homomorphism of rings and u is a unit in f, prove that f(u) is a unit in S

A: We recall the definition of a unit and then try to solve this problem. It is the definition of an…

Q: 11. Let R and R' be two rings. A mapping f: R→R' is called an antihomomorphism, if f(x+y)=f(x) +…

A: Let x, y ∈ R. Then (g)(x + y) = f(g(x + y)) = f(g(x) + g(y)) = f(g(x)) + f(g(y)) =(fg)(x)+(fg)(y),…

Q: Let p: Q[x] → M2(R) be defined by p(ao + a1 + a2x² + ...+ anx") = ( ) ao (1) Write down o(2 +x –…

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