My problem is: If the sum of to non-units in a ring $R$ is non-unit, then the Jacobson radical $J(R)$ is maximal. I need help please. I have no idea how to start. I thought that if the set of all non-units is an ideal of $R$, then it turns to be the unique maximal left ideal of $R$ and so it is $J(R)$.
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Can you show that this implies the set of non-units is an ideal? – Angina Seng Dec 24 '18 at 21:52