True or False 1. If f: R→S is a ring isomorphism, then ker f = {0} where 0 is the zero element in R. 2. By judging the operation tables for 2ℤ/8ℤ, we can say that this quotient ring is isomorphic to ℤ_4. 3. If R=ℤ_12 and I=(3), then R/I is a field. 4. Let P be a proper ideal of a ring R and a,b be any elements of R. If ab∈P and a∉P implies that b∈P, then P is a prime ideal of R. 5. The ring ℚ[x] of polynomials with coefficients from the field of rational

True or False 1. If f: R→S is a ring isomorphism, then ker f = {0} where 0 is the zero element in R. 2. By judging the operation tables for 2ℤ/8ℤ, we can say that this quotient ring is isomorphic to ℤ_4. 3. If R=ℤ_12 and I=(3), then R/I is a field. 4. Let P be a proper ideal of a ring R and a,b be any elements of R. If ab∈P and a∉P implies that b∈P, then P is a prime ideal of R. 5. The ring ℚ[x] of polynomials with coefficients from the field of rational

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Number 22 a and b
9. Suppose that (R,+,.) be a commutative ring with identity and x E rad R, then (а) (х) — R (b) 1 — хER" (c) (1 + x) ± R (d) No Choice (a) O (b) (c) (d)
Do both parts please Don't skip 2nd part
10. Suppose that (R, +,.) be a commutative ring with identity, then R/ rad R is .. (a) semi-simple (b) integral domain (c) field (d) No Choice
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…
5
please, just answer the last 3. Thank you
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
9. Iff js a homomorphism from a ring R into a ring S then show that f(0)=0 1 :: f(- a) = -f(a), V a e R.
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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