§ Handy characterization of adding an element into an ideal, proof that maximal ideal is prime
§ The characterization
Let be an ideal. The ideal generated by adding to is
defined as . We prove that .
§ Quotient based proof that maximal ideal is prime
An ideal is prime iff the quotient ring is an integral domain. An
ideal is maximal is a field. Every field is an integral domain,
I was dissatisfied with this proof, since it is not ideal theoretic: It argues
about the behaviour of the quotients. I then found this proof that argues
purly using ideals:
§ Ideal theoretic proof that maximal ideal is prime
Let be a maximal ideal. Let such that . We need
to prove that . If , the problem is done.
So, let . Build ideal . . Since
is maximal, . Hence, there are solutions for
Now, . ( by assumption).
let be a maximal ideal. let such that . we need
to prove that .
if , then the problem is done. so, let . consider
the ideal generated by adding into . .
We have shown that . Hence, .
Also, , \implies . Therefore,
. Since is maximal, this means that
Therefore, . Hence, there exists some such