(February 2013) Suppose that \(R\) is commutative and has a multiplicative identity \(1\). Let \(I\subseteq J\subseteq R\) be ideals, and suppose that the quotient ring \(R/I\) is a field. If \(I\subsetneq J\), prove that \(1\in J\). [In fact, it is a Theorem from Math 350 that \(J=R\) in this case, but you are only being asked to prove that \(1\in J\). In particular, however, you may not quote the \(J=R\) theorem.]
Solution
Proof. BecauseI ⊊J, there existsr∈J rI. By the coset relation, r /∈I impliesI +r⁄=I +0 = 0 R/I. Thus,I +r is nonzero. By the definition of field, every nonzero element of R/I has a multiplicative inverse. Hence, there exists I +s∈R/I such that (I +r)(I +s) =I + 1. So I +rs =I + 1, and therefore 1−rs∈I⊆J by the coset relation. Note that rs∈J, because J is an ideal of R and r∈J. Thus, since J is closed under addition, we have 1 = (1−rs) +rs∈J. QED