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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.

| | | | | created: 2019-04-11 22:46:27 | modified: 2019-08-04 06:08:18 | by: bookofproofs | references: [1272], [8152], [8189]

1.Proof: (related to "Addition, Subtraction and Multiplication of Congruences, the Commutative Ring $\mathbb Z_m$")

2.Corollary: Sums, Products, and Powers Of Congruences

Edit or AddNotationAxiomatic Method

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Bibliography (further reading)

[8152] Jones G., Jones M.: “Elementary Number Theory (Undergraduate Series)”, Springer, 1998

[8189] Kraetzel, E.: “Studienbücherei Zahlentheorie”, VEB Deutscher Verlag der Wissenschaften, 1981

[1272] Landau, Edmund: “Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie”, S. Hirzel, Leipzig, 1927