Definition:Ring (Abstract Algebra)/Ring Less Zero

Definition

Let $\struct {R, *, \circ}$ be a ring whose zero is $0$.

It is convenient to have a symbol for $R \setminus \set 0$, that is, the set of all elements of the ring without the zero.

Thus we usually use:

$R_{\ne 0} = R \setminus \set 0$

Also denoted as

Most sources use $R^*$ to denote the set of elements of a ring without the zero.

However, this is also used by some sources to mean the group of units of such a ring $R$.

Hence $\mathsf{Pr} \infty \mathsf{fWiki}$ prefers the more explicit notation $\R_{\ne 0}$.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $21$. Rings and Integral Domains