Tag: Series: Constructing Compatible Operations

Recently in my ring theory class I heard about the theorem that if a ring satisfies for some constant , then it must be commutative. Now, note that this statement is made entirely in terms of multiplication. Thus, going off of the theme discussed in my post The Unreasonable Effectiveness of Definitions in Mathematics, we can wonder: is this in fact true for all monoids? If not, then what “data” is needed in the monoid to ensure that we can construct a compatible addition operation turning it into a ring? For such a monoid, we would be able to conclude commutativity.