Monoid ring

In abstract algebra, a monoid ring is a ring constructed from a ring and a monoid, just as a group ring is constructed from a ring and a group.

Let R be a ring and let G be a monoid. The monoid ring or monoid algebra of G over R, denoted R[G] or RG, is the set of formal sums ${\displaystyle \sum _{g\in G}r_{g}g}$, where ${\displaystyle r_{g}\in R}$ for each ${\displaystyle g\in G}$ and r_g = 0 for all but finitely many g, equipped with coefficient-wise addition, and the multiplication in which the elements of R commute with the elements of G. More formally, R[G] is the set of functions φ: G → R such that {g : φ(g) ≠ 0} is finite, equipped with addition of functions, and with multiplication defined by

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