3. We have seen in lectures that, for any integer n > 0, there is a ring homomorphismYn: Z → Zn taking a to the remainder on dividing a by n. Show that the map: Z→ Z2 × Z3,taking a € Z to (42(a), 3(a)), is a ring homomorphism. Find the kernel and imageof , and use the First Isomorphism Theorem to deduce thatZ/(6) ≈ Z₂ X Z3.

Here given that ring homomorphism , Here for each is onto .Now another defined map is Here we have…

3. We have seen in lectures that, for any integer n > 0, there is a ring homomorphism Pn Z→ Zn taking a to the remainder on dividing a by n. Show that the map : : ZZ2 x Z3, taking a € Z to (p2(a), 3(a)), is a ring homomorphism. Find the kernel and image of , and use the First Isomorphism Theorem to deduce that Z/(6) Z₂ x Z3.

3. We have seen in lectures that, for any integer n > 0, there is a ring homomorphism Pn Z→ Zn taking a to the remainder on dividing a by n. Show that the map : : ZZ2 x Z3, taking a € Z to (p2(a), 3(a)), is a ring homomorphism. Find the kernel and image of , and use the First Isomorphism Theorem to deduce that Z/(6) Z₂ x Z3.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: Construct a ring homomorphism Z|V2| → Z/(7). Verify it is indeed a ring homomorphism (show all…

Q: 1- Which of the following is a homomorphism? a)O f:Z→Z, f(x) = x+2 %3D b) O f: Z12 → Z30, ƒ(x) = 20x…

A: Consider a function .Then f is a homomorphism if and only if . .

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: 5. Assume that R = a, b, e, d e Z and R [a] (b) a, b, , Given that 0:R+ R defined by 0( D- is a…

Q: Let I = = {[x] x, y = R} and J = {[2] ZER} E Consider the ring homomorphism y: I → R defined as (a)…

Q: Let R be a ring and f : R→ R be defined by f(x) = x3. Check All that are correct. O fis a ring…

A: As per bartleby guidelines for more than three sub parts in a question only first three should be…

Q: 3. Let f: RS be a ring homomorphism (a) If I is an ideal of S, prove the pre-image of I, that is…

A: Let I be an ideal of a ring R , and Let f:R→S be a ring homomorphism. To prove: f-1I is an ideal of…

Q: Denote by Zm = {[0]m, [1]m, ..., [m1]m} the ring of integers modulo m. Consider the rings R = Z24…

A: To decide whether a given map θ is an isomorphism of earrings, we should take a look at if it…

Q: Exercise 1. Let R and S be rings with identity, p: R → S a homomorphism of rings with p(1r) = 1s, s…

A: let R and S be rings with identity, ρ:R→S a homomorphism of rings with ρ1R=1S , s∈S and f∈Rx…

Q: Let A e M2(Q). 2 Define a ring homomorphism ф: Q] —> М2(Q), f(x) → f(A). (i) Show that A² – 2A – 3 =…

Q: Let : R → S be a ring homomorphism. Now let M be a subring of R and N a subring of S. Prove that…

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

Q: Problem 4. Let R = = consider the map: a b {(86) | a, b, c € Z}. Prove that R is a ring. Then 0 C o:…

Q: Let R be a ring with identity 18 Show that the map Ø: ZRgive by Ø(n) = nx1R. is a ring homomorphism.…

A: Consider the provided question,

Q: Consider ϕ: ℤ × ℤ → ℤ defined by ϕ(a,b) = a-b. Given that ϕ is a homomorphism, find its kernel and…

A: Let ϕ:*(Z*Z)->Z is a function. Then ϕ is a homomorphism iff ϕ(a+b)=ϕ(a)+ϕ(b) for all a,b ∈Z*Z

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]…

Q: 3. Are the rings 2Z, 3Z isomorphic. Any isomorphism is linear for addi- tion, so it has the form f…

A: Ring Isomorphism: Letus consider two rings R and S.Let us consider the mapping f: R→S. Then it is…

Q: Let R be a ring and f : R → R be defined by f(x) = x4. Check All that are correct. O fis not onto…

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

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: W 3. Lat R.S.T by rings and let w 4₁ iRDS and I₂ S&T boring. homomorphisms. Show that! V bu b)…

Q: Suppose R is commutative. Prove that the map oa : R[x] → R where $₁(p(x)) = p(a) for some a € R is a…

Q: (a) Let R and R' be two rings and : R→ R' a map. Define what it means for to be a ring homomorphism.…

Q: Let Suppose that A α V B B 4 C A' B' C' is a commutative diagram of groups and that the rows are…

Q: Construct a nontrivial homomorphism $: Zg → Zı5 >

Q: = {[0]m [1]m, [m - 1]m} the ring of integers modulo m. Z4 XZ6. Let : RS be the map Denote by Zm…

A: To summarize:The image of (\theta) is the set of all pairs (([x]_4, [y]_6)) where ([y]_6) can be…

Q: Suppose that : R Sand : S→T are ring homomorphisms. Recall that the composition of the functions and…

Q: 7. −ydx + (x + √xy)dy = 0

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

Q: Let f: Z6→ Z2 × Z3 be defined by the following specification of its values 0 → (0, 0) , 1 → (1, 1) ,…

Q: 14. Find all units of the ring C(R) = {ƒ : R → Rf is continuous}, where the operations are the usual…

A: A unit is an element which has multiplicative inverse. Using this definition we find the units in…

Q: R→s be a ring homomorphism. ve that if K is a subring of then R f(K) is a subring of S ve that c is…

A: Given, if f:R→S be a ring homomorphism. (i) To prove that if K is a subring of R then f(K) is a…

Q: Let :Z₁₂ →Z, be the homomorphism such that (1) = 2. (a) State the First Isomorphism Theorem. (b)…

Q: (a) Define a homomorphism : Zx Z→ Z with Ker = ((0, 1)). Show complete details. (b) Verify that is…

A: Given: We are given the following homomorphism: , and the subgroup Objective: The objective is to…

Q: If f: Z→ Z is the map defined by f(x)=2x, Is f a ring homomorphism when Z has its usual ring…

A: To determine if the map f: Z→Z defined by f(x)=2x is a ring homomorphism when Z has its usual ring…

Q: 5. Let K = Q(V2, i). Find all isomorphisms of K onto a subfield Q as extensions of the identity…

A: (5) Given, K=Q2, i Let φ:K→K, So, φ depends on φ2 and φi Now, minimal polynomial of 2 is x2-2, so,…

Q: - {(: ) -ahcer} (2) Let T : a, b, c ER} and g: T → R be defined by a = a

Question

3. We have seen in lectures that, for any integer n > 0, there is a ring homomorphism
Yn: Z → Zn taking a to the remainder on dividing a by n. Show that the map
: Z→ Z2 × Z3,
taking a € Z to (42(a), 3(a)), is a ring homomorphism. Find the kernel and image
of , and use the First Isomorphism Theorem to deduce that
Z/(6) ≈ Z₂ X Z3.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1: ring homomorphism

VIEW

Step 2: ring homomorphism

VIEW

Step 3: ring 1st Isomorphism theorem

VIEW

Solution

VIEW

Step by step

Solved in 4 steps with 32 images

SEE SOLUTION Check out a sample Q&A here

Follow-up Questions

Read through expert solutions to related follow-up questions below.

Follow-up Question

What is the kernel and image of the map

Solution

by Bartleby Expert

SEE SOLUTION

Similar questions

26 2. Let R = {a + bv2 | a, b e Z} and let R' consist of all 2 by 2 matrices of the form On the last homework, you showed R was a ring (in fact it is a subring of R). Show that R is a subring of Mat2(Z). Then show that p: R R' defined by a 26 p(a + bv2) a is an isomorphism.
15. (a) Show that S= {(a,a) | a e Z} is a subring of ZxZ. (Use the definition of addition and multiplication of direct products.) (b) Determine if T = {(a,-a) | ae Z} is a subring of ZxZ. 16. Let R be a commutative ring with unity and let
Let : R₁ → R₂ be a homomorphism of rings. Show that the mapping o': R₁[x] → R₂[x] given by o'(am +...+α₁x+αo): o(am) +...+ (a₁)x+ o(ao) is a homomorphism of rings with kero' = kero.
Let R be a ring. Consider the map Ø:Q[x]→Q defined by Ø(f(x))=f(3). Then, the Kernel of Ø is: * O None of the choices O O
Let R be a commutative ring with unity and let S be a nonempty subset of R. Show that I(S) := {f(x) = R[x] : f(a) = 0, for all a € S} is an ideal of R[x].
2. Assume that the set S I, y is a ring with respect to matrix addition and multiplication. D-r is a ring homomorphism from (a) Verify that the mapping e : S Z defined by e R to Z. (b) Find ker 0. (c) Find S/kere. (d) Find an isomorphism from S/kere to Z.
Determine if each of the following maps is a ring homomorphism. (a) f : R → R defined by f(x) %3D -x 1 -2 = (6 1) a ( (b) g : M2(IR) → M2(R) defined 1
Let R he a ring and f: RR be defined by f(z) = 1". %3D Check Al that are correct. O tis a group homomorphism for (R= Z4, +). %3D fis a ring homomorphism for (R = Z4, t.). fis not onto when R = Z7. |3D O tis one-to-one when R= Zg. %3D tis not onto when R = Zz. %3D tis a group homomorphism for (R = Z2, +).
(19) Let om : Z → Zm be the ring homomorphism defined by om (a) remainder of division of a by m. (a) Show that m : Z[x] → Zm[x] defined by om (anx + + a₁x + ao) = om(an)x² + is a ring homomorphism onto Zm[x]. + om (a₁)x+om (ao) = (b) Show that if f(x) = Z[x] and ¯m (f(x)) = Zm[x] both have degree n and σm(f(x)) does not factor in Zm[x] into two polynomials of degree less than n, then f(x) is irreducible Q[x]. (c) Use the previous part to show that x³ + 17x + 36 is irreducible in Q[x].
Number 6: show at K is splitting over Q.