# 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.

## 1 Answer

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
• Alright. That is equivalent to not being a zero divisor. – Ken Duna Jun 21 '16 at 20:51
• So the "opposite" of zero divisor is being a regular element? I'm sorry if I do make confusion but English is not my primary language and I'm translating all the terms I find in my language from my book to English :) – user1365914 Jun 21 '16 at 20:52
• That is correct. Many times people refer to regular elements as "non-zero-divisors". – Ken Duna Jun 21 '16 at 20:53
• Today I Learned "regular" elements are called "non zero divisors" in English :). Thank you a lot, have a great day @Ken Duna ! – user1365914 Jun 21 '16 at 20:54