# Proposition: Addition, Subtraction and Multiplication of Congruences, the Commutative Ring $\mathbb Z_m$

Let $a,b$ be integers and $m > 0$ be a positive integer. Then the addition "$+$", subtraction "$-$", and multiplication "$\cdot$" operations of the congruence classes $a(m), b(m)\in\mathbb Z_m$ are well-defined, setting

$$\begin{array}{rcl} a(m)+b(m)&:=&(a+b)(m),\\ a(m)-b(m)&:=&(a-b)(m),\\ a(m)\cdot b(m)&:=&(a\cdot b)(m).\\ \end{array}$$

and applying the addition, subtraction and multiplication of integers. In particular, the algebraic structure $(\mathbb Z_m,\cdot,+)$ of ever complete residue system modulo $m$ is a commutative unit ring.

Proofs: 1 Corollaries: 1

Algorithms: 1
Examples: 2
Lemmas: 3
Proofs: 4 5 6 7
Propositions: 8 9

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### References

#### Bibliography

1. Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927
2. Jones G., Jones M.: "Elementary Number Theory (Undergraduate Series)", Springer, 1998
3. Kraetzel, E.: "Studienbücherei Zahlentheorie", VEB Deutscher Verlag der Wissenschaften, 1981