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Commutative algebra is a subjects about commutative rings, ideals, modules, and related things. However, the name "commutative algebra" doesn't show any of these explicitly. If one do not know any background, it is quite likely to interpret this in a much narrower sense as algebras (in the sense of "Banach Algebra") that has a commutative binary product. "Commutative Ring theory" also seems to be a very straightforward term with meaning fairly close to "commutative algebra".

What is the motivation that leads us to use the term "commutative algebra"?

Shaun
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Ma Joad
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    It seems the role of the word commutative is clear to you. The word algebra does not refer only to the mathematical structure of "algebras", it is much much older. See for example https://en.wikipedia.org/wiki/Algebra#Etymology – s.harp May 11 '21 at 21:05
  • But it does "show them explicitly" if you interpret "algebra" as referring to the notion of an algebra $A$ over a ring $R$, i.e. a ring containing a central image of $R\ \ $ – Bill Dubuque May 11 '21 at 21:37

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