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, \(\displaystyle{Z}_{{20}}={\left\lbrace{0},{1},{2},….,{19}\right\rbrace}\)

\(2\times10=0\), Since \(\displaystyle{2}\ne{0},{10}\ne{0}\)

\(4\times4=0\), Since \(\displaystyle{4}\ne{0},{5}\ne{0}\)

\(4\times15=0\), Since \(\displaystyle{4}\ne{0},{15}\ne{0}\)

\(8\times5=0\), Since \(\displaystyle{8}\ne{0},{5}\ne{0}\)

\(12\times5=0\), Since \(\displaystyle{12}\ne{0},{5}\ne{0}\)

\(6\times10=0\), Since \(\displaystyle{6}\ne{0},{10}\ne{0}\)

\(8\times10=0\), Since \(\displaystyle{8}\ne{0},{10}\ne{0}\)

\(14\times10=0\), Since \(\displaystyle{14}\ne{0},{10}\ne{0}\)

\(16\times10=0\), Since \(\displaystyle{16}\ne{0},{10}\ne{0}\)

\(18\times10=0\), 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 \(1\times1=1\)

Units of \(3=7\), Since \(3\times7=1\)

Units of \(7=3\), Since \(7\times3=1\)

Units of \(9=9\), Since \(9\times9=1\)

Units of \(11=11\), Since \(11\times11=1\)

Units of \(13=17\), Since \(13\times17=1\)

Units of \(19=19\), Since \(19\times19=1\)

Hence, units are 1, 3, 7, 9, 11, 13, 17, 19.

These units cannot be zero-divisors.