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In the case of complete residue systems, the algebraic structure of congruences have been proved to be a commutative ring $\mathbb Z_m.$ If the integer $m$ was equal to a prime number $p$, then the ring became even a field, which was proven in this proposition. But what happens to the structure, if we consider reduced residue systems, i.e. if we allow only congruences classes co-prime to $m$? The following proposition clarifies this question.

## Proposition: Multiplicative Group Modulo an Integer $(\mathbb Z_m^*,\cdot)$

Let $m > 1$ be an integer. The algebraic structure $(\mathbb Z_m^*,\cdot)$ containing the congruence classes of a reduced residue systems modulo $m$ builds with respect to the multiplication operation $”\cdot”$ a commutative group called the multiplicative group modulo $m$.

### Notes

• The star in the notation $\mathbb Z_m^*$ means that $0\not\in\mathbb Z_m^*,$ since otherwise we would have $m\in\mathbb Z_m^*,$ in contradiction to its elements being co-prime to $m.$
• For this reason, the multiplicative group $\mathbb Z_p^*$ is not to be confused with the field $\mathbb Z_p$ if $p$ is a prime number.
• Similarly, an “addition” operation is not defined in $\mathbb Z_m^*,$ like it was the case in the commutative ring $\mathbb Z_m$ or in the field $\mathbb Z_p.$

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