4. In Z[z], let be the ideal generated by z and 4 (that is, the set ofall elements of the form 4a + zg(x))(a) Show 14(b) Prove < 2,4> is not a principal ideal, that is, #for any h(z) E Z[x]

Given: x,4 is an ideal generated by x and 4. To show : 1∉x,4 and x,4 is not a principal ideal.

Answered: 4. In 2[z], let < 2,4> be the ideal…

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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This question is of ring theory
(1) Let I be a proper ideal of the commutative ring R with identity. Then I is a ........ if and only if the quotient ring R/I is a field. (i) prime ideal (ii) primary ideal (iii) Maximal ideal (2) Z/(5): (i) (5) (ii) 5Z (iii) {[0],[1],[2],[3],[4]} (3) If R is an integral domain has non zero characteristic, then Char(R)=..... (i) 5 (ii) 4 (iii)9 (4) Let K be integer ring module 12 and let I=([4]) and J=([6]) be ideals of K. Then [2] belong to ... (i)I+J (ii)I.J (iii) I.J+ I (5) If f(x) then f(x) is reducible in Z5[x]. (i) x³ + 2x² + 2x + 1 (ii) x² + 1 (iii) x² + 2
Do both parts please Don't skip 2nd part
2
8
3. Determine the Jordan Canonical Form of A if the characteristic polynomial of A is given by (1-A)²(2-x)³ with dim(Ker(A-I)) = 1 and dim(Ker(A- 21)) = 2.
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10. Consider the irreducible polynomial f = X' + X³ + X² + X + 1 in Z[X]. Let a = [X] in the ring Z[X]/(f). Express (a + 1)(a² + 1) in the form aza + aza² + aa + ao where a; E Z.
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This question is of Ring theory
(7) In the ring (Z, +,.), we get n {P:P non trival prime ideal in Z} = ...... (а) ф (b) (Z, +, .) (c) ({0}, +,.) (d) No Choice (a) (b) (c) (d)
448_1

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4. In 2[z], let < 2,4> be the ideal generated by z and 4 (that is, the set of all elements of the form 4a + zg(x)) (a) Show 1 <1,4> (b) Prove < 2,4> is not a principal ideal, that is, <1,4> for any h(z) E Z[x]