Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

If there are, Are there unity (but not division) rings of this kind? Are there non-unity rings of this kind?

Sorry, I forgot writting the non division condition.

share|cite|improve this question
why not? Take the "polynomials" with integer coefficients in two non-commuting variables $x$ and $y$. – André Nicolas Nov 5 '13 at 3:54
@anon:Ah, yes, I was only thinking of multiplication. My mistake. Let me delete that; it was a stupid comment. – user99680 Nov 5 '13 at 4:01
up vote 4 down vote accepted

Take the "polynomials" with integer coefficients in two non-commuting variables $x$ and $y$. If you don't want a unit, use even integers only.

A related example replaces integer coefficients by coefficients in $\mathbb{Z}_2$.

share|cite|improve this answer
... or the ideal of all nonconstant polynomials. – Hurkyl Nov 5 '13 at 4:00
Do you know anything about the first (corrected) question? – Carlos Pinzón Nov 5 '13 at 4:29
My example happens not to be a division ring, so it happens to answer the revised question. The quaternion answers can be modified to be a non-division ring, by using the integer quaternions $a+bi+cj+dk$ where $a,b,c,d$ are integers. – André Nicolas Nov 5 '13 at 4:32

Yes, in fact much more can be said: There are rings with $1$ such that every non-zero element has a multiplicative inverse. These rings are called division rings or skew-fields.

The real quaternions are an example of a non-commutative division ring.

share|cite|improve this answer

Yes. In fact not only can we force every nonzero element to not be a zero divisor, we can force every nonzero element to be invertible. You get a division ring (also called skew field).

Perhaps one of the most famous historical examples: the quaternions. They are defined by

$$\Bbb H=\{a+bi+bj+ck:i^2=j^2=k^2=ijk=-1,a,b,c,d\in\Bbb R\}.$$

As for nonunital rings: simply take a unital ring with no zero divisors at hand (say $\Bbb H$) and look at a non-unital subring. For example the elements $2\Bbb Z+2i\Bbb Z+2j\Bbb Z+2k\Bbb Z\subseteq\Bbb H$.

share|cite|improve this answer

Another kind of example: non-commutative deformations of commutative integral domains, e.g. $\mathbb C[x,y]$ with the commutation relation $[x,y] = 1$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.