Rings

Rings

A ring $(R, +, \cdot)$ is a set $R$ with two binary operations satisfying:

1. $(R, +)$ is an abelian group
2. Multiplication is associative: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$
3. Distributive laws hold:
   • $a \cdot (b + c) = a \cdot b + a \cdot c$
   • $(a + b) \cdot c = a \cdot c + b \cdot c$

If multiplication is commutative, we call it a commutative ring. If there is a multiplicative identity $1$, we call it a ring with unity.

Example: $\mathbb{Z}_n$

The integers modulo $n$ form a commutative ring with unity under addition and multiplication mod $n$.

def Zn_add(a, b, n):
    return (a + b) % n

def Zn_mul(a, b, n):
    return (a * b) % n

Zero Divisors

An element $a \neq 0$ is a zero divisor if there exists $b \neq 0$ such that $a \cdot b = 0$.

In $\mathbb{Z}_6$: $2 \cdot 3 = 0$, so both 2 and 3 are zero divisors.

In $\mathbb{Z}_5$ (prime modulus): there are no zero divisors — this makes it an integral domain.

Units

An element $a$ is a unit (invertible) if there exists $b$ such that $a \cdot b = 1 \pmod{n}$. The units form a group under multiplication, denoted $\mathbb{Z}_n^*$.

Your Task

Implement zero_divisors(n) returning the sorted list of zero divisors in $\mathbb{Z}_n$, and units(n) returning the sorted list of units (multiplicatively invertible elements).