Definition 1

A local ring is a nonzero ring that has the equivalent properties of Proposition 1.

A ring A is local if and only if the opposite ring $A^o$ is local.

If A is a local ring, then the set $\mathfrak{r}$ of noninvertible elements of A is a two-sided ideal of A; it contains every left or right ideal of A not equal to A. The ring $A/\mathfrak{r}$ is therefore a field, which we call the residue field of A. The set $\mathfrak{r}$ is the unique maximal left (resp. right, two-sided) ideal of A; we simply say that $\mathfrak{r}$ is the maximal ideal of A.