Prove that it's not possible to have a ring homomorphism to .
The kernel of the ring homomorphism 0 : Z18 + Z6 given by °([2]18) = [x]6 is: List all distinct ideals in the ring Q of rational numbers: List all distinct principal ideals of the ring Z6:
5. Let I be an ideal in a ring R. Prove that the natural ring homomorphism T: RRI has kernel equal to I.
(3.) (a) Suppose that y: R S is a ring homomorphism. Please prove that (-a) = -f(a) for all a ER (b) Suppose R and S are rings. Define the zero function y: R S by pa) = Os for all GER. Is y a ring homomorphism? Please explain. (4.) Suppose that p is a prime number and 4: Z, Z, is defined by wa) = a.
18. Let o: R+ S be a ring homomorphism. Prove each of the following statements. (a) If R is a commutative ring, then (R) is a commutative ring. (b) (0)=0. (c) Let 18 and 1s be the identities for R and S, respectively. If o is onto, then (1r) = 1s. (d) If R is a field and $(R) +0, then (R) is a field.
= a (mod n) is a ring homomorphism. (10) Suppose that o Z Z defined by ¢(a) (a) (5 Pts) Prove that o is injective. Answer (b) (5 Pts) Prove that o is surjective onto its image. Answer = a (mod n) is a ring homomorphism. (10) Suppose that o Z Z defined by ¢(a) (a) (5 Pts) Prove that o is injective. Answer (b) (5 Pts) Prove that o is surjective onto its image. Answer
Algebraic Structures 7. Please show that every homomorphism image of ring is a ring.
Please solve all questions 1. Let 0 : Z/9Z+Z/12Z be the map 6(x + 9Z) = 4.+ 12Z (a) Prove that o is a ring homomorphism. Note: You must first show that o is well-defined (b) Is o injective? explain (c) Is o surjective? explain 2. In Z, let I = (3) and J = (18). Show that the group I/J is isomorphic to the group Z6 but that the ring I/J is not ring-isomorphic to the ring Z6. 3....
Question 5 (6 points) Let o: R+S be a ring homomorphism. Suppose that o(R) and ker o contain no nonzero nilpotent elements. Prove that contains no nonzero nilpotent elements.
need answer as soon as possible. thanks Consider the ring Rix) of polynomials with real coefficients, with operations polynomial addition and polynomial multiplication (you don't have to prove this is a ring). For example, for the polynomials f(x)=1+2x+3x2 and g(x)=3-5x, we have f(x)+g(x)= (1+2x+3x2)+(3-5x)-4-3x+3x2 and f(x)g(x)(1+2x+3x2)(3-5x)=3+X-X2-15x). Show that the function h: RIX-R given by h(f(x)=f(0) is a ring homomorphism. Then describe the kernel ker(h).
(Abstract Algebra-Ring Theory) In the quotient ring Z2[x]/(z6 + 1), verify that the ideal consisting of all multiples of g(x) = x4 +x2 + 1 contains all polynomials of the form a +baaz2 + ba3 + az4 (6,2) triple redundancy code bx (the corresponding codewords form the In the quotient ring Z2[x]/(z6 + 1), verify that the ideal consisting of all multiples of g(x) = x4 +x2 + 1 contains all polynomials of the form a +baaz2 + ba3 +...