 Solution

ID: past-exam-of-the-algebra-analysis-and-differential-equations-2-course-of-the-university-of-oxford/2004/3/i/solution

An ideal is an additive subgroup

I \subseteq R

with

r I \subseteq I

for all

r \in R

. A ring homomorphism preserves addition and multiplication; for the rings with identity here use the standard unital convention

θ (1_{R}) = 1_{S}

. If

a, b \in ker θ

, then

θ (a - b) = 0

, and if

r \in R

then

θ (r a) = θ (r) θ (a) = 0

. Thus the kernel is an ideal.

I \neq = R

such that an ideal

J

satisfying

I \subseteq J \subseteq R

is either

I

R

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