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An elementary result in ring theory is that if $R$ is a commutative ring with unity and $M$ is a maximal ideal of $R$, then $R/M$ is a field. There are many proofs of this, as you can see here.

But my question is, are there any constructive proofs of this result? That is, given a ring $R$ and a maximal ideal $M$ of $R$, is there a procedure to calculate the multiplicative inverse of any given element of $R/M$?

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In such generality there is no constructive proof. However, if $R$ happens to be an Euclidean ring then $M$ is principal (say it is generated by $m$), computing the inverse of $a\in R$ (which is not in $M$) is possible using a Bezout relation between $a$ and $m$. That is you have to do an extended Euclidean algorithm.

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