4. Let y: R→ S be a ring homomorphism. Prove that ': R[X] → S[X] given by soʻ (ao + a1X + ..a,X") = 9(a0) +y(a1)X + ... + p(a,)X" is a ring homomorphism.

Q: Let fg,:I→R2 begivenby f(x) = (0, x), g(x) = (1, x). a. Specify a homotopy h:I × I→R2 from f to g.…

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Q: 4. Prove that the function that sends an ordered pair (p,q) of elements in P(R) to * p(x) q(x)e da…

A: Please refer to the explanation.Explanation:Step 1:Step 2:Step 3:Step 4:Step 5:

Q: 2. Consider the following functions. Are these ring homomorphisms? If yes, prove it. If no, provide…

A: a) f:Z→Z given by f(x) = 3x For this function to be a ring homomorphism, it needs to satisfy two…

Q: 7. a) Let R be an integral domain and a € R. Show that the function f : R[x] → R given by f(p(x)) =…

A: 7) a) Let   be an integral domain and .        Let   be given by  ,         To  show  that   is  a…

Q: The mapping : (Z6, +6) → (U4, X4) on my avong bas 0 is arounico at wey song has 18 uniton ai n3"…

A: Mapping :

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: Let : R→ R' be a ring homomrphism of R onto R' with kernel K. If A' is a subring of R', let A {a €…

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Q: 17. Let R, R', R" be rings and 0₁: R→ R' and 02: R'→R" be homomorphisms. Prove that 0201: RR" is a…

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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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Q: 4. Show that the function : R² × R²: → R given by =u²[¹7¹] inner product on R². v does not define an

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Q: congruence class d'+mZ has order d.

A: To prove that the congruence class has order , we need to show two things:raised to the power of is…

Q: (†) \u,v≤U ([u‡ 0 and v‡0] → u • v‡ 0); (⋆) Vx,y,z€U ([z ‡ 0 and zx = zy] →x=y).

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Q: Consider X = [1,2]? with the relation (x1, y1)~(x2, Y2) if x1Y2 = x2y1. Also define 3x g: X → [1,2]…

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