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:

Ring Axiom $\text A$: Addition forms an Abelian Group

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:

Ring Axiom $\text M0$: Closure under Product:

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

Ring Axiom $\text M1$: Associativity of Product:

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

Ring Axiom $\text M2$: Identity Element for Ring Product:

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

Ring Axiom $\text C$: Commutativity of Ring Product:

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

Then:

Ring Axiom $\text D$: Distributivity of Product over Addition:

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

$\blacksquare$

Also see

• Quotient Epimorphism from Integers by Principal Ideal

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 $2$
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): congruence (modulo $n$)