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.