# Terminology: Groups, rings, fields, etc.

Groups, semigroups, fields, rings, integral domains, vector spaces, R-modules... these are all approximately the same sort of "stuff", but each one refers to a slightly different combination of required properties. Is there a general term that collectively refers to these types of "stuff"? Also, which branch of mathematics do such objects live in (broadly speaking)?

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Algebraic structures for the first question, and Abstract Algebra for the latter. – J. W. Perry Nov 19 '13 at 18:42

They go so far as to define $n$-ary operations on sets. Many different theorems (for example, the basic isomorphism theorems) which are usually proven separately for all those objects are proven all at once for large classes of algebraic objects in the eyes of universal algebra.