Integers Modulo m Commutative Ring with Unity

Theorem

For all $m \in \N: m \ge 2$, the algebraic structure $\left({\Z_m, +_m, \times_m}\right)$ is a commutative ring with unity $\left[\!\left[{1}\right]\!\right]_m$.

The zero of $\left({\Z_m, +_m, \times_m}\right)$ is $\left[\!\left[{1}\right]\!\right]_m$.

Proof

First we check the ring axioms:

• A: The Additive Group of Integers Modulo m $\left({\Z_m, +_m}\right)$ is a group.
• M0: The Multiplicative Monoid of Integers Modulo m $\left({\Z_m, \times_m}\right)$ is closed.
• M1: The Multiplicative Monoid of Integers Modulo m $\left({\Z_m, \times_m}\right)$ is associative.
• D: $\times_m$ distributes over $+_m$ in $\Z_m$.

Then we note that Multiplicative Monoid of Integers Modulo m $\left({\Z_m, \times_m}\right)$ is commutative.

Then we note that the Multiplicative Monoid of Integers Modulo m $\left({\Z_m, \times_m}\right)$ has an identity $\left[\!\left[{1}\right]\!\right]_m$.

Finally we note that $\left[\!\left[{0}\right]\!\right]_m$ is the identity of the additive group $\left({\Z_m, +_m}\right)$.

$\blacksquare$

Also see

• Canonical Epimorphism from Integers by Principal Ideal

Sources

• Iain T. Adamson: Introduction to Field Theory (1964)... (previous)... (next): $\S 1.1$: Example $4$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 21$: Example $21.1$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 24$
• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 1.6$: Theorem $5$
• B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970): $\S 1.2$: Ring Example $2$