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.
Best Answer
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