Ring of Integers Modulo m is Ring

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:

$A$: 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:

$M0$: $\struct {\Z_m, \times_m}$ is closed.

$M1$: $\struct {\Z_m, \times_m}$ is associative.

$M2$: $\struct {\Z_m, \times_m}$ has an identity $\eqclass 1 m$.

$C$: $\struct {\Z_m, \times_m}$ is commutative.

Then:

$D$: $\times_m$ distributes over $+_m$ in $\Z_m$.

$\blacksquare$

Also see

• Canonical Epimorphism from Integers by Principal Ideal

Sources

• 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): $\S 1.1$: Example $4$
• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): $\S 1.6$: Theorem $5$
• 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 1.2$: Some examples of rings: Ring Example $2$