(March 2005) Let \(F\) be a field and let \(J\subseteq F\) be an ideal of \(F\). Prove that either \(J=\{0\}\) or \(J=F\).
Solution
Proof. If J ={0}, then we are done. If J⁄={0}, then there is a∈J with a⁄= 0. Since F is a field, a has a multipllicative inverse a−1. Then 1 = a−1a∈J since J is an ideal. But an ideal that contains 1 is the whole ring, so J =F . QED
Maximal Ideals. Know the definition of a maximal ideal M⊆R. Criteria for R/M to be a Field. Let R be a commutative ring with unit. Know that ideal M⊆ R is maximal if and only if R/M is a field. Here is a problem that asks you to prove part of this result.