What's the difference between $(\mathbb Z_n,+)$ and $(\mathbb Z_n,*)$?

Question

I noticed that $(\mathbb Z_n,+)$ and $(\mathbb Z_n,*)$ are not the same thing. For example 2 is not invertible in $(\mathbb Z_6,*)$. However, since $\bar2+\bar4=\bar0$, thus it is invertible in $(\mathbb Z_6,+)$.

I found the following problems:

1. Am I right? If so, why does any book not mention this difference which is very important?
2. When we study groups, what's the standard operation of $\mathbb Z_n$ when the operation is omitted?
3. When we study rings, do we generally use $(\mathbb Z_n,+)$ to be the abelian group of the ring $(\mathbb Z_n,+,*)$?
4. If I work with $(\mathbb Z_n,+)$ I would lose these theorems which I proved for $(\mathbb Z_n,*)$?

• An element $a$ in $(\mathbb Z_n,*)$ is an unit iff $a$ and $n$ are coprimes
• An element $a$ in $(\mathbb Z_n,*)$ is an zero divisor iff $a$ and $n$ aren't coprimes

I'm sure there are a lot of student with this same doubt.

I really need help.

Thanks a lot.

Answer 1

1) Usually, $\,(\Bbb Z_n\,,\,+)\;$ would represent an additive cyclic group, which elements are usually represented as $\,\{\overline0\,,\,\overline 1\,\ldots\,\overline{n-1}\}\;$ or as $\,\{0,1,2,...,n-1\}\pmod n\,$ .

The set $\,(\Bbb Z_n\,,\,+\,,\,\cdot)\,$ represents the ring of residues modulo $\,n\,$ , with the same addition as above and multiplication modulo $\,n\,$ .

The set of invertible elements in the above ring is usually denoted by $\,\Bbb Z_n^*\;$, and sometimes $\,(\Bbb Z_n^*,\cdot)\;$ , and it is the set of elements in $\,\{0,1,2,...,n-1\}\pmod n\,$ which are coprime with $\,n\,$ (and non-zero, to avoid problems). This is a group under the usual multiplication in the above ring.

Of course, what you prove for a set under some operation may, and may not, be true for the same set, or a subset, under a different operation. Perhaps it'll be good you tell us what things were you thinking about.

Answer 2

Well, $(Z_{n}, *)$ is just the set $Z_{n}$ and a binary operation multiplication modulo $n$. Yes, you are right, this is entirely different from $(Z_{n}, +)$. The reason the books will not mention it because the former is not a group and hence not discussed, but nobody will say that they are the same things.

Answer 3

$(\mathbb Z_n,+)$ is cyclic. It is a regular polygon in the euclidian plane.

$(\mathbb Z_n,*)$ is a much complicated thing that makes the wonders of great mathematicians (Pythagoras, Gauss, Euler,...).