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

Chapter5: Rings, Integral Domains, And Fields

Section5.1: Definition Of A Ring

Problem 20E

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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
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:…
Qa: Prove that the ring of rational numbers (Q, +,.) is division ring
please explain.
7. Let R be a commutative ring and let F = F(R, R) be the set of functions f : R R. Functions in F can be added and multiplied pointwise using the operations from R, i.e. define (S + 9)(x) = f(x) + g(x) (f9)(2) = f(a)g(x) (a) Prove F is a commutative ring. (b) Let I denote the identity function 1(r) = r in F. Prove that the map p : R[X] → F given by p(a, X" +. + a1 X + ao) = a," + . + a1l + ao ... is a ring homomorphism, where R[X] is the ring of polynomials with coefficients in R. 1 of 2 MATH 4107: Homework 10 (c) Give an example showing that p is in general not injective (hint: try R= Z/2Z). Find ker(p) when R = Z/pZ where p is prime. Note that the fact that p is not injective means that one cannot in general view polynomials in R[X] as R-valued functions on R, which is in sharp contrast to the case of Z[X], R[X], or C[X].
true or false ?

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