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Semigroup Ring

A semigroup ring R[S] is a ring of finite formal sums sum_(s in S)a_ss indexed by a semigroup S, with coefficients a_s in a ring R. Addition is coefficientwise, and multiplication is obtained by extending the multiplication of S and the multiplication of R distributively, so

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

(Gilmer and Parker 1974). If R is a unit ring, the underlying R-module is a free module with basis S. If S is a group, this construction is the group ring. If S is the semigroup of n-tuples of nonnegative integers under componentwise 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.Gilmer, R. and Parker, T. "Divisibility Properties in Semigroup Rings." Mich. Math. J. 21, 65-86, 1974. https://doi.org/10.1307/mmj/1029001210.

Cite this as:

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

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