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Semigroup Ring

A semigroup ring R[S] is the free R-module with basis a semigroup S, equipped with multiplication obtained by extending the multiplication of S and the multiplication of the ring R bilinearly. Thus its elements are finite formal sums sum_(s in S)a_ss, and

(sum_(s in S)a_ss)(sum_(t in S)b_tt)=sum_(s,t in S)a_sb_t(st).

If S is a group, this construction is the group ring. If S=N^n under addition, then R[S] is the polynomial ring R[x_1,...,x_n].

See also

Group Ring, Monoid, Polynomial Ring, Semigroup

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References

Gilmer, R. Commutative Semigroup Rings. Chicago, IL: University of Chicago Press, 1984.

Cite this as:

Weisstein, Eric W. "Semigroup Ring." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SemigroupRing.html

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