Quotient Epimorphism from Integers by Principal Ideal
Theorem
Let $m$ be a strictly positive integer.
Let $\ideal m$ be the principal ideal of $\Z$ generated by $m$.
The restriction to $\N_m$ of the quotient epimorphism $q_m$ from the ring $\struct {\Z, +, \times}$ onto $\struct {\Z, +, \times} / \ideal m$ is an isomorphism from the ring $\struct {\N_m, +_m, \times_m}$ of integers modulo $m$ onto the quotient ring $\struct {\Z, +, \times} / \ideal m$.
In particular, $\struct {\Z, +, \times} / \ideal m$ has $m$ elements.
Proof
Let $x, y \in \N_m$.
By the Division Theorem:
\(\ds \exists q, r \in \Z: \, \) | \(\ds x + y\) | \(=\) | \(\ds m q + r\) | for $0 \le r < m$ | ||||||||||
\(\ds \exists p, s \in \Z: \, \) | \(\ds x y\) | \(=\) | \(\ds m p + s\) | for $0 \le s < m$ |
Then $x +_m y = r$ and $x \times_m y = s$, so:
\(\ds \map {q_m} {x +_m y}\) | \(=\) | \(\ds \map {q_m} r\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map {q_m} {m q} + \map {q_m} r\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map {q_m} {m q + r}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map {q_m} {x + y}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map {q_m} x + \map {q_m} y\) |
and similarly:
- $\map {q_m} {x \times_m y} = \map {q_m} {x y} = \map {q_m} x \map {q_m} y$
So the restriction of $q_m$ to $\N_m$ is a homomorphism from $\struct {\N_m, +_m, \times_m}$ into $\struct {\Z / \ideal m, +_{\ideal m}, \times_{\ideal m} }$.
Let $a \in \Z$.
Then:
- $\exists q, r \in \Z: a = q m + r: 0 \le r < m$
so:
- $\map {q_m} a = \map {q_m} r \in q_m \sqbrk {\N_m}$
Therefore:
- $\Z / \ideal m = q_m \sqbrk \Z = q_m \sqbrk {\N_m}$
Therefore the restriction of $q_m$ to $\N_m$ is surjective.
If $0 < r < m$, then $r \notin \ideal m$ and thus $\map {q_m} r \ne 0$.
Thus the kernel of the restriction of $q_m$ to $\N_m$ contains only zero.
Therefore by the Quotient Theorem for Group Epimorphisms, the restriction of $q_m$ to $\N_m$ is an isomorphism from $\N_m$ to $\Z / \ideal m$.
Work In Progress In particular: Reference is made throughout to $\struct {\N_m, +_m, \times_m}$. It needs to be shown that this is the same (at least up to isomorphism) as the ring of integers modulo $m$ $\struct {\Z, +_m, \times_m}$.) You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by completing it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{WIP}} from the code. |
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $24$. The Division Algorithm: Theorem $24.4$