Ring of Integers Modulo m is Ring
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Theorem
For all $m \in \N: m \ge 2$, the ring of integers modulo $m$:
- $\struct {\Z_m, +_m, \times_m}$
is a commutative ring with unity $\eqclass 1 m$.
The zero of $\struct {\Z_m, +_m, \times_m}$ is $\eqclass 0 m$.
Proof
First we check the ring axioms:
- $\text A$: The Integers Modulo $m$ under Addition form Abelian Group.
- From Modulo Addition has Identity, $\eqclass 0 m$ is the identity of the additive group $\struct {\Z_m, +_m}$.
From Integers Modulo m under Multiplication form Commutative Monoid:
- $\text M 0$: $\struct {\Z_m, \times_m}$ is closed.
- $\text M $: $\struct {\Z_m, \times_m}$ is associative.
- $\text M 2$: $\struct {\Z_m, \times_m}$ has an identity $\eqclass 1 m$.
- $\text C$: $\struct {\Z_m, \times_m}$ is commutative.
Then:
- $\text D$: $\times_m$ distributes over $+_m$ in $\Z_m$.
$\blacksquare$
Also see
Sources
- 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 1$. Rings and Fields: Example $4$
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 6$. The Residue Classes: Theorem $5$
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $2$: Some examples of rings: Ring Example $2$