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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.

Update: There are some great examples below. It would be good for some folks to chime in about the following basic structures to round out the list.

• Euclidean domains
• Principle ideal domains
• Unique factorization domains
• Modules of various sorts
• Finite fields

3 added 289 characters in body

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.

Update: There are some great examples below. It would be good for some folks to chime in about the following basic structures to round out the list.

• Euclidean domains
• Principle ideal domains
• Unique factorization domains
• Modules of various sorts
• Finite fields
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