List all zero-divisors in Z_20. Can you see relationship between the zero-divisors of Z_20 and the units of Z_20?

Assume that R be a commutative ring and a be a nonzero element of R.
Zero-divisors An element a of a ring R is called a zero divisor if there exists a nonzero x such that ax = 0.
From the definition of zero divisors, find the zero divisors of ${\displaystyle Z_{20}}$ in the following.
Since, ${\displaystyle Z_{20}=\{0,1,2,\dots .,19\}}$
$2\times 10=0$, Since ${\displaystyle 2\ne 0,10\ne 0}$
$4\times 4=0$, Since ${\displaystyle 4\ne 0,5\ne 0}$
$4\times 15=0$, Since ${\displaystyle 4\ne 0,15\ne 0}$
$8\times 5=0$, Since ${\displaystyle 8\ne 0,5\ne 0}$
$12\times 5=0$, Since ${\displaystyle 12\ne 0,5\ne 0}$
$6\times 10=0$, Since ${\displaystyle 6\ne 0,10\ne 0}$
$8\times 10=0$, Since ${\displaystyle 8\ne 0,10\ne 0}$
$14\times 10=0$, Since ${\displaystyle 14\ne 0,10\ne 0}$
$16\times 10=0$, Since ${\displaystyle 16\ne 0,10\ne 0}$
$18\times 10=0$, Since ${\displaystyle 18\ne 0,10\ne 0}$
Therefore, zero divisors of ${\displaystyle Z_{20}}$ are 2, 4, 5, 6, 8, 10, 12, 14, 15, 16 and 18.
A unit in a ring is an element u such that there exists ${\displaystyle u^{-1}}$ where ${\displaystyle u.u^{-1}=1}$
Now find the units of ${\displaystyle Z_{20}}$ in the following.
Since the elements which are relatively prime to 20 is called units.
Therefore, the relatively primes to 20 are 1, 3, 7, 9, 11, 13, 17, and 19.
Then,
Units of $1=1$, Since $1\times 1=1$
Units of $3=7$, Since $3\times 7=1$
Units of $7=3$, Since $7\times 3=1$
Units of $9=9$, Since $9\times 9=1$
Units of $11=11$, Since $11\times 11=1$
Units of $13=17$, Since $13\times 17=1$
Units of $19=19$, Since $19\times 19=1$
Hence, units are 1, 3, 7, 9, 11, 13, 17, 19.
These units cannot be zero-divisors.