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I am teaching an intro to ring theory, and after grading the first quiz, I realize most of my students are under the assumption that rings must be commutative. I have given them the example of matrices over the reals, but clearly we need to spend a little more time on non-commutative rings.

What are the most basic examples of non-commutative rings? (Most basic in the sense that I don't want to spend all day proving that this thing is indeed a ring.) Specifically, are there interesting quotients of $M_2(\mathbb{R})$?

Please note: this is a rings-first abstract algebra class, so the group ring of a non-abelian group will take us a little far afield.

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    $\begingroup$ $M_2(\mathbb{R})$ doesn't have any non-trivial quotients, but the ring of upper triangular matrices is an interesting subring. $\endgroup$
    – Dag Oskar Madsen
    Oct 14, 2017 at 19:11

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The quaternion ring is a pretty simple example of a non-commutative ring (a skew-field, even).

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Why not turn $M_2(\mathbb{R})$ into a multiplication on $\mathbb{R}^4$? Here's a fun, somewhat weird, ring: $$ \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] \left[ \begin{array}{cc} x & y \\ z & w \end{array}\right] = \left[ \begin{array}{cc} ax+bz & ay+bw \\ cx+dz & cy+dw \end{array}\right]$$ hence $$ (a,b,c,d) \star (x,y,z,w) = (ax+bz, ay+bw, cx+dz, cy+dw)$$ So, $(\mathbb{R}, \star, +)$ gives a fairly unfamilar ring which is a fairly concrete example. Addition is as usual, but, multiplication is a bit surprising. The unity here is $(1,0,0,1)$. Of course, you see right through that, but, to students this would be shrouded in mystery for a little while.

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  • $\begingroup$ Oh nice, and we get to build an isomorphism, too. $\endgroup$
    – David Steinberg
    Oct 14, 2017 at 20:25
  • $\begingroup$ The unity is $(1, 0, 0, 1)$, right? $\endgroup$
    – Eli Rose
    Oct 15, 2017 at 1:44
  • $\begingroup$ @EliRose Thanks! Fixed. $\endgroup$
    – James S. Cook
    Oct 15, 2017 at 4:26
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How about $\mathbb{Z}[i]$ together with conjugation, thought of as a ring element? Conjugation and multiplication by $i$ do not commute.

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    $\begingroup$ I've never thought about this ring, so let me ask a naive question: As an abelian group, is this free and generated by 1, $i$, and $c$, where $c$ is conjugation? $\endgroup$
    – David Steinberg
    Oct 14, 2017 at 20:15
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    $\begingroup$ @DavidSteinberg No, you also have $ic$ which corresponds to swapping the real and imaginary axes. Really, it's isomorphic to $M_2(\mathbb{Z})$. $\endgroup$
    – Adam
    Oct 14, 2017 at 20:39
  • $\begingroup$ Oh right, of course! Thank you $\endgroup$
    – David Steinberg
    Oct 14, 2017 at 20:42
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    $\begingroup$ @Adam I don't see how this ring can be isomorphic to $M_2(\mathbb Z)$ since its only idempotents are $0,1$ (if $(z+cw)^2=z+cw$ with $z,w\in\mathbb Z[i]$ then $(z+\bar z-1)w=0$, but $z+\bar z$ is even so $w=0$). Via the natural action on $\mathbb Z[i]$, it is the subring of $M_2(\mathbb Z)$ containing matrices whose main and anti diagonals have even sum. $\endgroup$
    – stewbasic
    Oct 16, 2017 at 4:56
  • $\begingroup$ Whoops! @stewbasic is correct, the ring above is a proper subring of $M_2(\mathbb{Z})$. It seems I forgot I wasn't working over the reals/rationals. $\endgroup$
    – Adam
    Oct 16, 2017 at 15:05
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The Weyl algebra (the ring of differential operators with polynomial coefficients). It admits a quite concrete description and there are many interesting things to say about it.

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Another elementary ring to consider is the Exterior Algebra of a vector space. That has the advantage of having both commutative and non-commutative parts, and having a relationship to linear algebra via determinants.

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Since it's an intro to rings course, all the students will be familiar with functions. Also, they should have the sense already that function composition is non-commutative. So how about an example of a ring of functions where the addition is just the usual $+$ and the multiplication is given by composition?

Something like $\operatorname{End}(G)$, the ring of endomorphisms of an abelian group $G$ will work. Or if you don't want to talk about groups at all, you can still use $\operatorname{End}(G)$ for a specific $G$ that students will be familiar with.

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Group rings.

If the students are already familiar with some non-commutative groups (e.g. the symmetric groups) and you take a field to go with it that is familiar to the students (e.g. $\mathbb{R}$ or $\mathbb{F}_2$), then these rings should be rather natural and the multiplication and addition easy to understand.

Furthermore, you can choose any group and any ring to start with, so you won't run out of examples that soon. :)

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Related to your example of $M_2(\mathbb{R})$ is to consider the ring $UT_2(\mathbb{R})$ of upper triangular $2 \times 2$ matrices.

An interesting nontrivial quotient is the map $UT_2(\mathbb{R}) \to \mathbb{R} \times \mathbb{R}$ which sends a matrix to the pair of entries on its diagonal.

The kernel $I$ of this quotient is the ideal of all matrices whose only entry is in the top-right corner. This is a particularly interesting ideal since $I^2 = 0$.

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