Homework 6
Regen Torrens
Section 6.1 Number 25
Let f:RS be a homomorphism of rings and let K = {r є R | f(r) = 0s}. Prove that K is an ideal in R.
Theorem: A nonempty subset I of a ring R is an idea if and only if it has these properties:
i)
ii)
If a, b є I, then a-b є I;
If r є R and a є I, then ra є I and ar є I.
Let a, b є K, then f(a) = 0 and f(b) = 0, so f(a-b) = f(a)-f(b) = 0-0 = 0, therefore a-b є K
K is nonempty because 0r є K, since f(0s) = 0s. Let r є R and a є K, then f(a) = 0, so f(ar) = f(a)f(r) =
f(r)0 = 0, so ar є K, similarly f(ra) = f(r)f(a) = f(r)0 = 0, so ra є K.