Question

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

Commutative Algebra
ANSWERED
asked 2020-11-30
List all zero-divisors in \(\displaystyle{Z}_{{20}}\). Can you see relationship between the zero-divisors of \(\displaystyle{Z}_{{20}}\) and the units of \(\displaystyle{Z}_{{20}}\)?

Answers (1)

2020-12-01

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.

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