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" (cf. Lurie).
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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See also
References
 Laplaza, M. Coherence for distributivity. Coherence in categories, 2965. Lecture Notes in Mathematics 281, SpringerVerlag, 1972.
 Lurie, J. Derived Algebraic Geometry V: Structured Spaces