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A commutative $k$-algebra (with $k$ a field or at least a commutative ring) is an associative unital algebra over $k$ such that the multiplicative operation is commutative?. Equivalently, it is a commutative ring $R$ equipped with a ring homomorphism $k \to R$.
There is a generalization of commutativity when applied to finitary monads in $Set$, that is generalized rings, as studied in Durov's thesis.
Commutative algebra is the subject studying commutative algebras. It is closely related and it is the main algebraic foundation of algebraic geometry. Some of the well-known classical theorems of commutative algebra are the Hilbert basis theorem and Nullstellensatz and Krull's theorem?, as well as many results pertaining to syzygies, resultants and discriminants.
Michael Atiyah, I. G. Macdonald, Introduction to commutative algebra, 1969, 1994
H. Matsumura, Commutative algebra, 2 vols.; see also the online summary notes by D. Murfet, Matsumura.pdf, Matsumura-Part2.pdf
D. Eisenbud, Commutative algebra: with a view toward algebraic geometry, Grad. Texts in Math. 150, Springer-Verlag 1995.
James Milne, A primer of commutative algebra, (online notes in progress) webpage, pdf
Discussion of commutative algebra with constructive methods:
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