Let I = {[xx, y = R} and J = {[²] 1z € R}EConsider the ring homomorphism : I→ R defined as(a) Show that is a ring homomorphism.(b) Use FIT for rings to show that I/JR as rings.([*])= x - y.

Answered: Image /qna-images/answer/140f8140-d233-4b8a-9e39-fd8a217d246d.jpg

Answered: Let I = = {[x] x, y = R} and J = {[2]…

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

Similar questions

2. Determine whether or not the set is an ideal in the ring M2x2 (R). J = 00 {( 9) | TER} 0 r
(k) Let f: A B be a mapping of sets, A a system of subsets of A, and B a system of subsets of B. Define S(A) = {S(X) C B : X e A). (8) = {r-)eA :Y e B}. Prove that f-1(B) is a ring if B is a ring. (1) Show that f(A) is not necessarily a ring if A is a ring. (m) Show that -(B) is a a-algebra if B is a o-algebra. (n) Prove that R(s-(8) = (R(B). where R(C) is the minimal ring containing e.
3. Consider the ring R[x], and its ideal I = (x²). Then the quotient ring R[x]/I provides an abstract construction of the ring R[ɛ]. (a) Define a function : R[ɛ] → R[x]/I by (a + b) = a + bx + I Verify that is a ring homomorphism. (b) Consider a polynomial f(x) = a + bx + c2x² + €3x³ + ….. +Cnx² € R[x]. Show that f(x) + I = a + bx + I are equal as equivalence classes. (c) Using part (b), explain why the homomorphism from part (a) is surjective.
If f: Z→ Z is the map defined by f(x)=2x, Is f a ring homomorphism when Z has its usual ring operations? How would you prove that? If not, what could be a counter-example?
3. Determine whether the following mappings are ring homomorphisms: f: Q- Q defined by f(x)= |x| for all x e Q. a) b) c) g: CM₂ (R) defined by g (a + bi) = a b -b a h: Z[√2] → Z[√2] defined by h( a + b √2) =) a - b √2
Each of the quotient rings R/I on the left is isomorphic to one of the rings S on the right. Match each ring with its isomorphic partner, and prove that they really are isomorphic by constructing a surjective ring homomorphism p: RS with kernel I. R/I S A) Z[X]/(8, 12, X) 1. ZZ 2. Q 3. R B) Q[X]/(X² - 2) 4. C 5. F4 C) R[X]/(X-√2) 6. F8 7. ZA 8. Zg D) R[X]/(x²+X+2) 9. {a+b√2 a,bQ} 10. a {( 8 b ) | ab € F₁ } a E) R[X]/(X²) F) Z₁₂[X]/(X + X²+1) a 11. | a, b € F8 0 a 12. {( o b) | ab € R } a 0 a In examples 5, 6, 10, and 11 the symbols F4 and Fg denote a field of 4 and a field of 8 elements respectively. In examples 10, 11, and 12 the ring operations are addition and multiplication of matrices. Some methods you might use to try and match up R/I and S: (i) guess, (ii) think about representatives in the quotient ring, and what the multiplication rules are and try and match that up with something in the S column, (ii) try and make up a homomorphism from R to somewhere that would have elements…
Q) Let (H,+10, -10) be a subring of the ring (Z10, +10, -10 ), where H = {[0], [2], [4], [6], [8]}. Then answer the following: 1) Fill this table 10 [0] [2] [4] [6] [8] [0] [2] [4] [6] [8] 2) Find all the idempotent elements of H. 3) Find all the nilpotent elements of H. of 1
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.
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.
5. Let R be a commutative ring with identity. Let f: R[x] - f(p(x)) = E, ai, where p(x) = ao+a1x + a2x² + ... + amx™m. Prove that f is a homomorphism of rings. Notice that f(p(x)) = p(1), where 1 = 1R E R. → R be defined by 4 4. Risc
Let R be a ring with unity e. Verify that the mapping θ: Z---------- R defined by θ (x) = x • e is a homomorphism

Recommended textbooks for you

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Mathematics For Machine Technology

Advanced Math

ISBN:

9781337798310

Author:

Peterson, John.

Publisher:

Cengage Learning,

Basic Technical Mathematics

Advanced Math

ISBN:

9780134437705

Author:

Washington

Publisher:

PEARSON

Topology

Advanced Math

ISBN:

9780134689517

Author:

Munkres, James R.

Publisher:

Pearson,

SEE MORE TEXTBOOKS

Let I = = {[x] x, y = R} and J = {[2] ZER} E Consider the ring homomorphism y: I → R defined as (a) Show that is a ring homomorphism. (b) Use FIT for rings to show that I/JR as rings. X ([ + ]) Y = x - y.