Please list some examples of common examples of algebraic structures. I was thinking answers of the following form.


"When I read about a [insert structure here], I immediately think of [example]."


Or maybe you think about a small number of examples. For example, when someone says "group", maybe you immediately think of one example of an abelian group and one example of a non-abelian group. Maybe you have a list of examples that you test new theorems on. 


I am an analyst by training. When I read algebra, I can follow the logic line by line, but I don't have the repertoire of examples in my head that I have for analysis and so it's hard for me to picture anything. 



Canonical examples of algebraic structures
Neat question...


Abelian Group: Z or Z/nZ


Nonabelian Group: Dihedral groups or GL\_n


Commutative Ring: Z or C[x]


Noncommutative Ring: Matrix rings


Division Ring: Hamilton's Quaternions


Field: R or C


Lie Algebra: sl\_2


