# Can a ring have no zero divisors while being non-commutative and having no unity?

I was wondering if, in a ring, the property of having no zero-divisors (except for zero itself) is independent from the ring being commutative or from having a unity (i.e.multiplicative identity) so I started looking for a ring with the following properties:

1. non-commutative
2. no unity (i.e. no multiplicative identity: a so-called "rng")
3. no zero-divisors

I came up with the set of 2 x 2 matrices with even entries: $$M_2(2\Bbb Z)$$ endowed with the usual matrix addition and matrix multiplication. It is:

1. non-commutative: $$\begin{pmatrix}2&2\\2&0\end{pmatrix}\begin{pmatrix}0&2\\2&2\end{pmatrix}\neq\begin{pmatrix}2&2\\2&0\end{pmatrix}\begin{pmatrix}0&2\\2&2\end{pmatrix}$$
2. no unity: $$\begin{pmatrix}1&0\\0&1\end{pmatrix}\notin M_2(2\Bbb Z)$$

But unfortunately it does have zero divisors: $$\begin{pmatrix}2&0\\0&0\end{pmatrix}\begin{pmatrix}0&0\\0&2\end{pmatrix}=\begin{pmatrix}0&0\\0&0\end{pmatrix}$$

So, can you come up with a ring having those three properties? Or a proof that such a group cannot exist?

• A 'rng' is not a noncommutative ring; it is a ring without unity. – Servaes Jul 22 at 12:29
• Also; such a ring cannot be finite because in a finite ring, every element is either a unit or a zero divisor. – Servaes Jul 22 at 12:32
• @Servaes ah you're right! Just edited my post. – UndefinedBehavior Jul 22 at 13:26

Let $$R$$ be the ring of polynomials with integer coefficients in two non-commuting variables $$x,y$$, and let $$I$$ be the ideal generated by $$\{x,y\}$$.

Then $$I$$, regarded as a ring, satisfies all your specified conditions.

• So basically 𝐼 contains all the polynomials in 𝑅 with degree >=1 plus the 0 polynomial (additive identity). Agree? – UndefinedBehavior Jul 25 at 13:46
• Not exactly. The ideal $I$ consists of all polynomials in $R$ with constant term equal to zero. – quasi Jul 25 at 13:50
• That's what I meant. Sorry for my misleading wording. I meant polynomials not containing any monomial of degree 0. Also... if i'm not mistaken, the quotient ring is (isomorphic to) ℤ – UndefinedBehavior Jul 25 at 14:22
• Yes, that's right. – quasi Jul 25 at 14:27

One example of such a ring is the free algebra on two elements over the rng $$2\Bbb{Z}$$.

I think the example of a nontrivial ideal of the free algebra on two noncommuting variables is already an optimally simple answer, but I'd like to offer a few other noncommutative, nonfield domains that would also work.

Any nontrivial ideal of the Hurwitz quaternions or Lipschitz quaternions would do.

Well, each semigroup $$(S,\circ)$$ with no unit element can be extended to a monoid $$(S\cup\{e\},\circ,e)$$ which has a unit element $$e$$ not belonging to $$S$$. This is a general construction.

So the question whether the ring (i.e., multiplicative semigroup) has a unit element or not is more or less pointless.

• -1 This is a good comment but not a good answer. – Servaes Jul 22 at 12:28
• Well wrt my post it'd be more interesting if the opposite were true: i.e. if from any given monoid you could just remove the identity and get a bona-fide semi-group. Unfortunately this is not the case, you need to add the constraint that the monoid you start with must not be a group. – UndefinedBehavior Jul 22 at 13:24
• Moreover, while it's true that you ca easily extend a (multiplicative) semigroup with an identity element, if it is a ring you need to also define addition for the new element. – UndefinedBehavior Jul 22 at 14:24