# Is there a procedure to calculate the multiplicative inverse in a quotient by a maximal ideal?

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 $$R/M$$?

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.