Monoidring

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

Definition

Let be a commutative ring with unity and a monoid, then

With the addition

And the convolution

As a multiplication ring. The design is modeled after that of the polynomial ring. You write or just for the picture, which assumes the value and at the point otherwise. For example, is considered to

Has an identity element, namely, the identity element of and the neutral element of is.

If a group is the name of group ring or group algebra; the spelling is common.

Is via the algebra

Properties

• R [ G] is a commutative ring if and only if G is commutative as monoid.
• Each element can be written uniquely as with
• And are embedded in a natural way, namely by the injective or Monoidhomomorphismen ring and, wherein is as defined above.
• If G is a monoid and A, B commutative rings, a ring homomorphism, then there is a unique homomorphism. so that

Universal property

The Monoidring or Monoidalgebra can also - be defined by a universal property - up to isomorphism. And are as defined above. Denote the category of monoids and the category of ( associative ) algebras. Be the forgetful, that is, the functor that assigns to each algebra its multiplicative monoid.

Then the canonical embedding is universal, ie: If we do not have another monoid homomorphism into the multiplicative monoid of an algebra, then there exists a unique algebra homomorphism such that.

In the above construction of Monoidalgebra looks like this: .

If we denote with the functor which assigns to each monoid his Monoidalgebra over, so it is linksadjungiert to. So we get a very brief definition of Monoidalgebra, but you still have to prove the existence.

Examples

• R [ N0 ] is isomorphic to the polynomial ring in one indeterminate over R.
• More generally, if G is a free commutative monoid in n generators, then R [G ] is isomorphic to the polynomial ring in n indeterminates over R.

Generalizations

• It is a locally compact topological group. Is not discrete, the ring group contains no information on the topological structure. Therefore takes its role convolution algebra of integrable functions: it is a left invariant Hair measure on. Then, the space is formed with the folding

• Is a ring and a totally ordered group whose order is compatible with the group operation, ie