Corollary 4

The ring A is local if and only if the ring $A/\mathfrak{R}(A)$ is a field. If this is the case, then $\mathfrak{R}(A)$ is the set of noninvertible elements of A.

Denote the set of noninvertible elements of A by $\mathfrak{r}$. If the ring A is local, then its radical is equal to $\mathfrak{r}$ (VIII, p. 154, Example 1) and the ring $A/\mathfrak{r}$ is a field (VIII, p. 26). Conversely, suppose that the ring $A/\mathfrak{R}(A)$ is a field. By Corollary 2, we have $\mathfrak{r}=\mathfrak{R}$(A), so $\mathfrak{r}$ is a two-sided ideal of A. It follows that the ring A is local (VIII, p. 26, Definition 1).