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