2. (a) Let R and R' be two rings and ø: R→ R' a map. Define what it means for to be a ring homomorphism. (b) Let ø: Z → Z be given by ø(x)=2x. Show that is not a ring homomorphism. (c) Let R and R' be two rings and 6: R→ R' a map. Define what it means for to be a ring isomorphism. (d) Show that the rings 2Z and 3Z are not isomorphic. [1+2+1+2=6 marks]

a. If R is a commutative ring with unity, show that the characteristic of R[ x ] is the same as the characteristic of R. b. State the characteristic of Zn[ x ]. c. State the characteristic of Z[ x ].

Find Let f : R² → R² be an isomorphism where (C)--(4)-1] [8]

Prove. Let f : R → R' be a ring homomorphism. If I' is an ideal of R', then f^-1(I') is an ideal of R.

11. Let R and R' be two rings. A mapping f: R→R' is called an antihomomorphism, if f(x+y)=f(x) + f(y) and f(xy) = f(y)f(x) x, y € R. Let f, g be two antihomomorphisms of a ring R into R. Prove that fg: R R is a homomorphism.

(a) Let R and R' be two rings and ø: R → R' a map. Define what it means for ø to be a ring homomorphism. (b) Let ø: Z → Z be given by ø(n)=n². Show that ø is not a ring homomorphism. (c) Let R and R' be two rings and : R → R′ a map. Define what it means for to be a ring isomorphism. (d) Show that the ring Z is not isomorphic to the ring M. (Z). (e) Prove that Z, is a commutative ring with unity but it is not a field.

4) Let f : C → C be a homomorphism of rings with f(x) = x for all x E R. Show that f is either the identity or complex conjugation.

Which of the following is isomorphism? f:(Z, +) → (Z, +) where f(x) = 2x. f: (R, +) → (R*, .) where f(x) = 2*

6- Define f:Z- 42 by f(x) = 4x, for all x € Z. Then: (a) f is a ring epimorphism. (b) f is a ring homomorphism but it is not monomorphism. (c) f is not a ring homomorphism. (d) f is a ring homomorphism but it is not epimorphism.

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].