1

How can I describe the set of all zero divisors in the ring $\mathbb{Z} / b\mathbb{Z}$ for $b \in \mathbb{Z}, \, b\geq 2$ with a mathematical proof?

I know that the set of zero divisors just contains the equivalence classes of the numbers $x,y$ such that $x*y = b$. So now $[x], [y]$ are in the set of zero divisors. But i would like to give a mathematical prove and a with a precise description of the set

Slyrack
  • 23
  • 7
  • 1
    Try a couple of examples, like $b=2,3,4,5$ and $6$ first. See if you spot a pattern. Once you suspect what the pattern is, you can start working on a proof. – Arthur Jun 20 '21 at 13:00
  • I know that the set of zero divisors just contains the equivalence classes of the numbers $x,y$ such that $x*y = b$. So now $[x], [y]$ are in the set of zero divisors. But i would like to give a mathematical prove and a with a precise description of the set – Slyrack Jun 20 '21 at 13:05
  • @Slyrack That's exactly the kind of information which we want the question post itself to contain. The downvote and close vote you have gotten is likely because that's lacking. – Arthur Jun 20 '21 at 13:07

2 Answers2

4

Your conjecture is wrong, I'm afraid. Consider $b=6$. Then $[4]$ is a zero divisor, because $[4][3]=[0]$, but there is no $y$ such that $4y=6$.

However, this should lead you to a more precise conjecture: indeed $4\cdot3=12=2b$, so…

You may want to prove that an element in $\mathbb{Z}/b\mathbb{Z}$ is either invertible or a zero-divisor (considering $[0]$ a zero-divisor). I believe you can classify the invertible elements, so to get the right conjecture for the zero-divisors. The proofs will be almost applying the definitions.

egreg
  • 238,574
  • So can i describe the set of all 0 divisors as ${[x] \in \mathbb{Z}/b\mathbb{Z} : 2 \leq x \leq b-1 ,and ,\exists c \in \mathbb{Z} , with ,xc = db ,for ,d \in \mathbb{Z} }$? – Slyrack Jun 20 '21 at 13:31
  • @Slyrack That's almost correct; it should be the set of $[x]$ such that there are $c,d\in\mathbb{Z}$ such that $cx=db$ (the limitation $2\le x\le b-1$ is redundant, albeit harmless). But there's a better and simpler way to say the same thing. – egreg Jun 20 '21 at 13:38
  • Thank you. What would be that way to do it even better and simpler? I thought how I did it would be the "natural" way? – Slyrack Jun 20 '21 at 13:42
  • @Slyrack What about the gcd? – egreg Jun 20 '21 at 13:43
  • Is it: ${[x]: gcd(x,b)>1}$? – Slyrack Jun 20 '21 at 13:48
  • @Slyrack That's it: now try and prove it. – egreg Jun 20 '21 at 13:53
  • My attempt to prove it would be to first give my first description of the set (what I think follows directly from the definition of this ring) and then from this one I can rewrite the set with the gcd as my second description of the set was. What do you think? – Slyrack Jun 20 '21 at 13:58
1

Question: "How can I describe the set of all zero divisors in the ring Z/bZ for b∈Z,b≥2 with a mathematical proof?"

Answer: Here is an approach using the Chinese Remainder Theorem where you get more information on the units: What are they?

In $\Bbb Z/m\Bbb Z,$ if $a$ is not invertible then it is a zero divisor

hm2020
  • 1