Monoid ring

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A monoid ring can be viewed as a generalization of a polynomial ring . The powers of the variables are replaced by elements from a monoid , which is precisely defined below.


Be a commutative ring with one and a monoid, then is

with the addition

and folding

a ring as a multiplication . The construction is based on that of the polynomial ring . One writes or simply for the illustration that assumes the value at that point and otherwise . For example, then applies

possesses one element, namely , wherein the element is from and the neutral element is from .

If a group is, it is called group ring or group algebra ; the spelling is also common.

becomes - algebra via


  • is a commutative ring if and only if is commutative as a monoid or is the zero ring .
  • Each element can be clearly written as with
  • If not the zero ring is, are , and in a natural way in embedded, namely by the injective ring or Monoidhomomorphismen and wherein is as defined above.
  • If the null ring is, then is isomorphic to the null ring
  • If there is a monoid, commutative rings and a ring homomorphism , then there is a unique homomorphism . so that

Universal property

The monoid ring or the monoid algebra can also - apart from isomorphism - be defined via a universal property . Be and as defined above. It denotes the category of monoids and the category of (associative) algebras. Be the forgetful functor , d. H. the functor that assigns each algebra to its multiplicative monoid.

Then the canonical embedding is universal , i.e. i.e., if we have another monoid homomorphism into the multiplicative monoid of an -algebra , then there is exactly one -algebra homomorphism such that .

In the above construction of Monoidalgebra looks like this: .

If we denote the functor, which assigns its monoidalgebra to every monoid , it is left adjoint to . This gives us a very brief definition of monoid algebra, but one still has to prove its existence.


  • is isomorphic to the polynomial ring in an indefinite over .
  • If, more generally, is a free commutative monoid in generators, then is isomorphic to the polynomial ring in indefinite over .

Special cases

  • Let it be a locally compact topological group . If not discrete , the group ring contains no information about the topological structure of . So his role is played by the convolution algebra of integrable functions: it is a linksinvariantes Hair measure on . Then the space forms with the fold
as a product a Banach algebra .
  • Is a ring and a totally ordered group whose order is compatible with the group operation, i.e. H.
out and follows
so be
with With the convolution as multiplication and the component-wise addition becomes a ring. If there is a body, then it is an oblique body . If, for example, is with the natural order, then the ring of formal Laurent series with coefficients in .


  • Serge Lang: Algebra, Graduate Texts in Mathematics, Revised Third Edition (Springer, 2002, ISBN 0-387-95385-X )