Suppose there is a commutative ring $R$, without any restriction. Now suppose $a,b \in R$. If $1-ab$ is a non-unit, why is there at least one maximal ideal $M$ that $1-ab \in M$?
2026-04-11 22:01:04.1775944864
For a commutative ring $R$, why does $1-ab$ being a non-unit leads to $1-ab \in M$ for some maximal ideal $M$?
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If an element is not a unit, the ideal it generates is not all of $R$. Now, every proper ideal is contained in a maximal ideal.