$\gcd \cdot \mathrm{lcm}$ for cyclic rings

Question

A cyclic ring is a ring (or rng) whose additive group is cyclic.

Two elements of a commutative ring are associates $(\sim)$ iff they divide each other.

An element $d$ of a commutative ring is a $\gcd(a,b)$ iff:

• $d$ is a common divisor of $a$ and $b$, and
• any common divisor of $a$ and $b$ divides $d$.

An element $m$ of a commutative ring is an $\mathrm{lcm}(a,b)$ iff:

• $m$ is a common multiple of $a$ and $b$, and
• $m$ divides any common multiple of $a$ and $b$.

It looks like the formula

$\gcd(a,b) \cdot \mathrm{lcm}(a,b) \sim a \cdot b$

works for any cyclic ring, even if there is no unity in it, as long as $\gcd(a,b)$ exists.

For example, in $2\mathbb Z_{12}$:

$\gcd(4,8) \sim 4$
$\mathrm{lcm}(4,8) \sim 4$
$\gcd(4,8) \cdot \mathrm{lcm}(4,8) \sim 4 \sim 4 \cdot 8$

There are proofs of the $\gcd \cdot \mathrm{lcm}$ formula for an integral domain:
Prove that $\gcd(M, N)\times \mbox{lcm}(M, N) = M \times N$. and
Transfer Between LCM, GCD for Rings?

How do we show it for an arbitrary cyclic ring?

In an infinite cyclic ring $k \mathbb Z$ a nonzero $\gcd$ exists if only $k = 1$ since any $\gcd$ must divide itself.

I need help with a finite cyclic ring $k \mathbb Z_{kn}$.

Answer 1

1. Definition. Prime-power cyclic ring is a finite cyclic ring (rng) $p^m \mathbb{Z}_{p^n}$,
   where $p$ is a prime number, and $0 \le m \lt n$;

2. $a \sim p^k, m \le k \le n$ for any element $a$ of a prime-power cyclic ring $p^m \mathbb{Z}_{p^n}$:

   • $p^k \mathbb{Z}_{p^n}, m \leq k \leq n$, are the only ideals of $p^m \mathbb{Z}_{p^n}$;

3. $\gcd(a, b) \cdot \mathrm{lcm}(a, b) \sim a \cdot b$ in a prime-power cyclic ring:

   • assuming $gcd(a, b)$ exists, $a \sim p^x, b \sim p^y, m \le x \le y \le n$;
   • then $gcd(a, b) \sim p^{x - m}, lcm(a, b) \sim p^{y + m}$;
   • $\gcd(a, b) \cdot \mathrm{lcm}(a, b) \sim p^{x - y} \cdot p^{y + m} = p^x \cdot p^y \sim a \cdot b$

4. If $\gcd(a, b) \cdot \mathrm{lcm}(a, b) \sim a \cdot b$ in two rings (rngs) $A$ and $B$,
   then $\gcd(a, b) \cdot \mathrm{lcm}(a, b) \sim a \cdot b$ in the ring $A \times B$:

   • $\gcd((a,b), (c,d)) = (\gcd(a,c), \gcd(b,d))$;
   • $\mathrm{lcm}((a,b), (c,d)) = (\mathrm{lcm}(a,c), \mathrm{lcm}(b,d))$;
   • $(a, b) \sim (c, d) \iff a \sim c \land b \sim d$.

5. Any finite cyclic ring (rng) $k \mathbb Z_{kn}$ with $k > 0, n > 1$ is a direct sum of prime-power cyclic rings.

Note: in this proof $a \sim b$ in an rng $R$ if $a$ and $b$ generate the same principal ideal $R \cdot a + \mathbb{Z}a$.