University of Wisconsin-Madison
University of Wisconsin-Madison
Mathematics Department

Math 541
Modern Algebra
A first course in Abstract Algebra
Lecturer: Arun Ram

Fall 2007

 

Homework 7: Due October 24, 2007

 

To grade: 4, 11, 15.

  1. Define homomorphism, kernel and image and give some examples.

  2. Let f:G→ H be a group homomorphism. Show that kerf is a normal subgroup of G.

  3. Let f:G→ H be a group homomorphism. Show that Gkerf ≃imf.

  4. Let f:G⟶H be a group homomorphism. Show that f is injective if and only if ker f ={1}.

  5. Let f:G⟶H be a group homomorphism. Show that f is surjective if and only if im f =H.

  6. Define ring, subgroup, coset, R/I and ideal and give some examples.

  7. Let I be a subgroup of a ring R. Show that the operation on R/I given by

    ( r1 +I ) ( r2 +I ) = r1 +r2 +I
    is well defined.

  8. Let I be a subgroup of a ring R. Show that the operation on R/I given by

    ( r1 +I ) ( r2 +I ) = r1 r2 +I
    is well defined then I is an ideal of R.

  9. Let I be a subgroup of a ring R. Show that if I is an ideal of R then the operation on R/I given by

    ( r1 +I ) ( r2 +I ) = r1 r2 +I
    is well defined.

  10. Let I be an ideal of a ring R. Show that R/I with operations given by

    ( r1 +I ) + ( r2 +I ) = ( r1 + r2 ) +I and ( r1 +I ) ( r2 +I ) = r1 r2 +I
    is a ring.

  11. Determine the ideals of ℤ.

  12. Let f:R⟶A be a ring homomorphism. Show that ker f is an ideal of R.

  13. Let f:R⟶A be a ring homomorphism. Show that im f is a subgroup of R.

  14. Let f:R⟶A be a ring homomorphism. Show that im f is an ideal of A.

  15. Let f:R→ A be a ring homomorphism. Show that Rker f ≃im f.

  16. Let f:R⟶A be a ring homomorphism. Show that f is injective if and only if ker f ={0}.

  17. Let f:R⟶A be a ring homomorphism. Show that f is surjective if and only if im f =A.