In mathematics, a categorical ring is, roughly, a category equipped with addition and multiplication. In other words, a categorical ring is obtained by replacing the underlying set of a ring by a category. For example, given a ring R, let C be a category whose objects are the elements of the set R and whose morphisms are only the identity morphisms. Then C is a categorical ring. But the point is that one can also consider the situation in which an element of R comes with a "nontrivial automorphism".^{[1]}
This line of generalization of a ring eventually leads to the notion of an E_{n}ring.
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Introduction to Higher Mathematics  Lecture 17: Rings and Fields

A Ring is Commutative iff (a  b)(a + b) = a^2  b^2 Proof

MathHistory22: Algebraic number theory and rings I
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References
 ^ Lurie, J. (2004). "V: Structured Spaces". Derived Algebraic Geometry (Thesis).
 Laplaza, M. (1972). "Coherence for distributivity". Coherence in categories. Lecture Notes in Mathematics. Vol. 281. SpringerVerlag. pp. 29–65. ISBN 9783540379584.
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