# Is it true that the only regular elements in $Z_m$ are invertible ones?

I have this doubt. In a unitary and commutative ring $$Z_m = \{[0]_m, [1]_m,\ ...\ ,\ [m - 1]_m\}$$

There are only two "kind" of elements: invertible and zero divisors. Is it true to say that the only regular elements are the invertible elements?

Thanks.

If by regular, you mean not a zero divisor, then yes it is true. This would follow by your observation that there are only two "kinds" of elements in $\mathbb{Z}_m$.
Generally this is not true. For instance in the polynomial ring $\mathbb{R}[x]$, $x$ is neither a zero divisor nor invertible.
• By regular I mean $a \in A: \forall b,c \in A, a*b = a*c \Longrightarrow b=c$ – user1365914 Jun 21 '16 at 20:47