Q1) If (R, +,.) is integral domain, then the characteristic of R is either zero or prime.====Q2) Let (P(X),A,n) be a ring, where X {1, 2, 3}. Answer the following. (i) Find the zeroelement 0 and the identity element 1 in the ring P(X). (ii) Find the additive inverse of theelement {1,3} in P(X). (iii) Find the characteristic of P(X). (iv) Find all the zero divisors.Q3) Any ring R is without zero divisor if and only if the cancellation laws for multiplication=ca where a 0 implies b = c.are satisfied, that is for a, b, c ER, ab = ac and ba=x, then Char(R) = 2.Q4) Let R be a ring with identity R. If for every x R it holds x²

Step 1: Step 2: Step 3: Step 4:

Answered: Q1) If (R, +,.) is integral domain,…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.1: Polynomials Over A Ring

Problem 17E

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18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .
15. Let and be elements of a ring. Prove that the equation has a unique solution.
Let R be a commutative ring with characteristic 2. Show that each of the following is true for all x,yR a. (x+y)2=x2+y2 b. (x+y)4=x4+y4
Q(i) isomorphic to which quotient ring
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
/5. A division ring is a ring with identity in which every nonzero element has a multi- plicative inverse. Assuming (R,+,) is a division ring, prove that (cent R,+,) forms a field.
number 4 please
(R, +, .) (N, +, .) (R, +, .) (Z, +, .) | (Z6, +, .) (M3x3(IR), +, .) Ring Has identity Commutative Every nonzero element has inverse Has no zero divisor Division ring Integral domain Field Char (R)
The ring Z is isomorphic to the ring 3Z True False The product of x + x3 + 2x² and 2x=+ 4x + 6 in z, [x]. None
Find the characteristic of the ring (Z/20Z) /(5Z/20Z)
Pls. answer no. 4 and 5 only.1. Write the simplest form of the splitting field of the polynomial in no.10 using only 2 generators (take the positive conjugate). Write k,l,m,n ∈ ℚ, then write Q(k+lsqrt(u),m+nsqrt(v)) as Q(sqrt(u),sqrt(v)).2. Find all the roots in ℂ of the polynomial x^4-4x^3+3x^2+14x+26 with one of its roots being 3+2i. List down the roots as in no.11.[-a+sqrt(a^2-4b)]/2,[-a-sqrt(a^2-4b)]/2,[-c+sqrt(c^2-4d)]/2,[-c-sqrt(c^2-4d)]/2So roots are 3+2i,root2,root3,root4If the roots is not a fraction then remove the square brackets!Simplify the roots as possible.3. Find a monic polynomial of least degree in ℝ[x] that has 4i-1 and -3 as roots.write the powers of x as x,x^2,x^3,x^4,.... e.g. x^5-2x^4-4x^3+3x^2+2x+14. Find a monic polynomial of least degree in ℝ[x] that has 1-i and 2i as roots.write the powers of x as x,x^2,x^3,x^4,.... e.g. x^5-2x^4-4x^3+3x^2+2x+15. Find the splitting field for x^2-2sqrt(2)x+3 over ℚ(sqrt(2)). Find the roots first then simplify.Note:…
(Q1)Find all idempotent elements and nilpotent elements of the following rings 2- (Z/(3),+.) + (C. +,.)

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