Find all the zero divisors in a ring

[evinda]

Hello! :D
I am given the following exercise:Find all the zero divisors in the ring $\mathbb{Z}_{20}$.
For each zero divisor $[a]$,find an element $ \neq [0]$ such that $[a]=[0]$.
That's what I did..Could you tell me if it is right??
Zero divisors at the ring $\mathbb{Z}_{20}$: $\{ [2], [4], [5], [6], [8], [10], [12], [14], [15], [16], [18]\}$
The couples are:
$([2],[10]),([10],[2])$
$([4],[5]),([5],[4])$
$([4],[10]),([10],[4])$
$([4],[15]),([15],[4])$
$([5],[8]),([8],[5])$
$([5],[12]),([12],[5])$
$([5],[16]),([16],[5])$
$([6],[10]),([10],[6])$
$([8],[10]),([10],[8])$
$([8],[15]),([15],[8])$
$([10],[12]),([12],[10])$
$([10],[14]),([14],[10])$
$([10],[16]),([16],[10])$
$([10],[18]),([18],[10])$
$([12],[15]),([15],[12])$
$([14],[15]),([15],[14])$
$([15],[16]),([16],[15])$

 

[I like Serena]

Hello! :D
I am given the following exercise:Find all the zero divisors in the ring $\mathbb{Z}_{20}$.
For each zero divisor $[a]$,find an element $ \neq [0]$ such that $[a]=[0]$.
That's what I did..Could you tell me if it is right??
Zero divisors at the ring $\mathbb{Z}_{20}$: $\{ [2], [4], [5], [6], [8], [10], [12], [14], [15], [16], [18]\}$
The couples are:
$([2],[10]),([10],[2])$
$([4],[5]),([5],[4])$
$([4],[10]),([10],[4])$
$([4],[15]),([15],[4])$
$([5],[8]),([8],[5])$
$([5],[12]),([12],[5])$
$([5],[16]),([16],[5])$
$([6],[10]),([10],[6])$
$([8],[10]),([10],[8])$
$([8],[15]),([15],[8])$
$([10],[12]),([12],[10])$
$([10],[14]),([14],[10])$
$([10],[16]),([16],[10])$
$([10],[18]),([18],[10])$
$([12],[15]),([15],[12])$
$([14],[15]),([15],[14])$
$([15],[16]),([16],[15])$

Yep. It is right! (Cool)

 

[evinda]

Yep. It is right! (Cool)

Great!Thanks a lot! (Giggle)

 

[Deveno]

In fact, it is not hard to see that in $\Bbb Z_{20}$ we have:

$[k]$ is a zero divisor if and only if $\text{gcd}(k,20) > 1$.

We also have the following fact (which is not true for rings in general, but IS true for cyclic rings):

$[k]$ is a zero divisor, or $[k]$ is a unit, and never both.

This turns out to be very useful when examining properties of integers in general (often, we "reduce mod $n$" and then "lift" what we have learned in $\Bbb Z_n$ to $\Bbb Z$).

 

[blue_raver22]

Hello! It seems like you have correctly identified all the zero divisors in the ring $\mathbb{Z}_{20}$. Your list and pairings look good to me. Keep in mind that there may be more than one possible pairing for each zero divisor, so your list may not be exhaustive. Also, remember that in a ring, the zero element is always a zero divisor, so you can include $[0]$ as a zero divisor as well. Great job!