Mathematics 228(Q1), Assignment 5
Due : Monday, February 26, 2007
Exercise 1.(10 marks) Given functions f:T→Rand g:T→S, their product f×g:T→R×Sis the
function defined by
(f×g)(t) = (f(t), g(t)), t ∈T.
Assuming fand gare homomorphisms of rings, show f×gis a homomorphism of rings. Furthermore, show
that f×gis injective if both fand gare injective.
Exercise 2.(10 marks) Let Z[√2] be the ring of exercise 2, assignment 4 and f:Z[√2] →Z[√2] the function
defined by
f(a+b√2) = a−b√2.
Show fis an isomorphism of Z[√2].
Exercise 3.(10 marks) Let Rbe the ring of all functions f:R→R. Given a∈R, the function a:R→R
defined by
a(f) = f(a), f ∈R.
is referred to as evaluation at a. Verify that ais a homomorphism of rings. Is ainjective ? surjective ? Be
sure to justify your answers.
Exercise 4.(10 marks) Let f:R→Sbe a homomorphism of rings. Show that if r∈Rthen for all n∈Z
f(nr) = nf (r).
(Hint : Recall definition of multiples and the techniques used to prove their elementary properties.)
Exercise 5.(10 marks) Given a homomorphism f:R→Sof rings, its kernel is the set
ker f={r∈R:f(r) = 0S}.
(a) Verify that ker fis a subring of R.
(b) Show that ker fenjoys the following additional property : if k∈ker fand r∈Rthen both rk and k r
lie in ker f.
Exercise 6.(10 marks) Recall, if f:R→Sand g:S→Tare functions then their composite g◦f:R→T
is the function defined by
(g◦f)(r) = g(f(r)), r ∈R.
(a) Assuming fand gare homomorphisms of rings, show g◦fis also a homomorphism.
(b) If fand gare isomorphisms, show g◦fis an isomorphism.
Exercise 7.(10 marks) Let f:R→Sbe an isomorphism of rings and let g:S→Rbe the inverse function
of f. Show gis an isomorphism of rings.(Hint : To verify g(s1+s2) = g(s1) + g(s2), consider its image under
the homomorphism fand use the facts that fis a homomorphism and f◦gis the identity map.)