# Trivial Rings and Zero-Divisor

I'm trying to do a homework question and I must have something basic messed up in my head. I've looked up Trivial Ring and I've looked up Zero-divisor. A trivial Ring is a Ring with the single element 0. A Zero-divisor is a nonzero x such that ax = xa = 0.

My question is the part that says "there is no nonzero element that when multiplied by 0 yields 0" because.. EVERY element when multiplied by 0 is 0, including 0.

When it says that a zero-divisor is a nonzero x such that ax = xa = 0, where are the x and a coming from? The x has to be nonzero and can come from anywhere, right? And a is from the ring in question? So if 0 can be a zero divisor for a non-trivial ring (a in the ring, x=0) then why can't it be a zero-divisor for the trivial ring? (a=the 0 in the ring, and x=0)?

Is it just saying that there is no nonzero element period, therefore you can't continue (leaving zero not a zero divisor)? That seems sloppy to me.

Can someone provide some clarification here?

• The use of zero divisor in the image is nonstandard: Usually one defines a zero-divisor of a ring $R$ to be a nonzero element $x$ of a ring for which there is a nonzero element $y$ such that $x y = 0$ or $y x = 0$. In particular, for the trivial ring there are no nonzero elements in the first place, and hence no zero divisors. – Travis Apr 13 '17 at 0:28
• @Travis Well, that is an overstatement. Perhaps it is just not the convention followed in your favorite books/disciplines. It's fair to say that this definition isn't universal certainly. – rschwieb Apr 13 '17 at 1:00
• Here's my completed HW question... dropbox.com/s/gifp9ova08bck6f/HW16.12a.png?dl=0 – Dana Hill Apr 13 '17 at 1:40
• @rschwieb Thank you for the clarification. I'm not an algebraist, but I am a working mathematician, and I'd never encountered another definition before. In fact, I checked several references before making that comment, to make sure I hadn't forgotten seeing some alternate definition somewhere. – Travis Apr 13 '17 at 11:46

Pay close attention to the definitions. A zero-divisor of a ring, $R$, is a nonzero element, $x \in R$ such that for some nonzero $a \in R$, $ax=xa=0$.

The trivial ring has no nonzero element, since there is only 0. In other words every element equals 0.

• On a nitpicky note, $a\in R$ should be nonzero as well. – Santana Afton Apr 13 '17 at 0:20
• Given how important it is to be accurate about definitions in this question, I wouldn't even call it a nitpick. – rschwieb Apr 13 '17 at 1:14

The convention followed in that passage is that any element $x$ (even zero) is considered a zero divisor only if there exists a nonzero element $y$ such that $xy=0$.

Is it just saying that there is no nonzero element period, therefore you can't continue (leaving zero not a zero divisor)?

Yes, apparently so. In the trivial ring, no such element can be found, so strictly speaking $0$ in the $0$ ring fails to fit that convention.

That seems sloppy to me.

If by sloppy you mean "totally consistent with itself" then OK. After all, the zero ring is a ring with identity, so $0$ is a unit... it would be weird to have a zero divisor that is a unit too!

Anyhow, the zero ring is a bit of a weird edge case that we typically stay away from. I'm not even a huge fan of the viewpoint in that passage, but one has to recognize the variety of consistent interpretations that exist.

where are the x and a coming from?

Well, from the ring of course. Where else? You are not given anything outside of the ring.

The moral of the story is just always to closely pay attention to the assumptions being used. It will vary from text to text, and sometimes even between your teachers.