Monoid ring

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.

definition

Be a commutative ring with one and a monoid, then is ${\displaystyle R}$ ${\displaystyle G}$

${\displaystyle R[G]:=\{\alpha \colon G\to R{\ {\big |}\ }\alpha (x)=0{\text{ für alle bis auf endlich viele }}x\}\,}$

with the addition

${\displaystyle (\alpha +\beta )(x):=\alpha (x)+\beta (x)}$

and folding

${\displaystyle (\alpha \beta )(z):=\sum _{xy=z}{\alpha (x)\beta (y)}}$

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 ${\displaystyle a\cdot x}$ ${\displaystyle ax}$ ${\displaystyle \alpha \in R[G]}$ ${\displaystyle x}$ ${\displaystyle a}$ ${\displaystyle 0}$

${\displaystyle (a\cdot x)(b\cdot y)=(ab)\cdot (xy)\quad {\text{für }}a,b\in R{\text{ und }}x,y\in G.}$

${\displaystyle R[G]}$possesses one element, namely , wherein the element is from and the neutral element is from . ${\displaystyle 1\cdot e}$ ${\displaystyle 1}$ ${\displaystyle R}$ ${\displaystyle e}$ ${\displaystyle G}$

If a group is, it is called group ring or group algebra ; the spelling is also common. ${\displaystyle G}$ ${\displaystyle R[G]}$ ${\displaystyle RG}$

${\displaystyle R[G]}$becomes - algebra via ${\displaystyle R}$ ${\displaystyle r\sum _{i}r_{i}g_{i}:=\sum _{i}rr_{i}g_{i}}$

properties

• ${\displaystyle R[G]}$is a commutative ring if and only if is commutative as a monoid or is the zero ring . ${\displaystyle G}$ ${\displaystyle R}$
• Each element can be clearly written as with ${\displaystyle \alpha \in R[G]}$ ${\displaystyle \alpha =\sum _{x\in G}a_{x}\cdot x}$ ${\displaystyle a_{x}:=\alpha (x)}$
• 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. ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle G}$ ${\displaystyle R[G]}$ ${\displaystyle f_{0}\colon R\to R[G],~f_{0}(r)=r\cdot e}$ ${\displaystyle f_{1}\colon G\to R[G],~f_{1}(x)=1\cdot x}$ ${\displaystyle 1\cdot x}$
• If the null ring is, then is isomorphic to the null ring ${\displaystyle R}$ ${\displaystyle R[G]}$
• If there is a monoid, commutative rings and a ring homomorphism , then there is a unique homomorphism . so that ${\displaystyle G}$ ${\displaystyle A,B}$ ${\displaystyle f\colon A\to B}$ ${\displaystyle h\colon A[G]\to B[G]}$ ${\displaystyle h\left(\sum _{x\in G}a_{x}x\right)=\sum _{x\in G}f(a_{x})x}$

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. ${\displaystyle G}$ ${\displaystyle R}$ ${\displaystyle \mathbf {Mon} }$ ${\displaystyle \mathbf {Alg_{R}} }$ ${\displaystyle R}$ ${\displaystyle U\colon \mathbf {Alg_{R}} \to \mathbf {Mon} }$ ${\displaystyle R}$

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 . ${\displaystyle \phi \colon G\to U(R[G]),g\mapsto 1g}$ ${\displaystyle f\colon G\to U(A)}$ ${\displaystyle R}$ ${\displaystyle A}$ ${\displaystyle R}$ ${\displaystyle {\bar {f}}\colon R[G]\to A}$ ${\displaystyle U({\bar {f}})\circ \phi =f}$

In the above construction of Monoidalgebra looks like this: . ${\displaystyle {\bar {f}}}$ ${\displaystyle {\bar {f}}\left(\sum _{i}r_{i}g_{i}\right)=\sum _{i}r_{i}f(g_{i})}$

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. ${\displaystyle R}$ ${\displaystyle F}$ ${\displaystyle F}$ ${\displaystyle U}$

Examples

• ${\displaystyle R[\mathbb {N} _{0}]}$is isomorphic to the polynomial ring in an indefinite over . ${\displaystyle R}$
• If, more generally, is a free commutative monoid in generators, then is isomorphic to the polynomial ring in indefinite over . ${\displaystyle G}$ ${\displaystyle n}$ ${\displaystyle R[G]}$ ${\displaystyle n}$ ${\displaystyle R}$

Special cases

See also : Group C * algebra

• 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 ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle \mathbb {C} [G]}$ ${\displaystyle G}$ ${\displaystyle \mu }$ ${\displaystyle G}$ ${\displaystyle L^{1}(G,\mu )}$

${\displaystyle (f*g)(\sigma )=\int _{G}f(\tau )g(\tau ^{-1}\sigma )\,\mathrm {d} \mu (\tau )}$

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. ${\displaystyle A}$ ${\displaystyle G}$

out and follows ${\displaystyle \alpha <\beta }$ ${\displaystyle \gamma <\delta }$ ${\displaystyle \alpha \gamma <\beta \delta ,}$

so be

${\displaystyle S(G,A)=\{f\colon G\to A\mid {\text{supp }}f{\text{ wohlgeordnet}}\}}$

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 . ${\displaystyle \mathrm {supp} \,f:=\{g\in G\mid f(g)\not =0\}.}$ ${\displaystyle S(G,A)}$ ${\displaystyle A}$ ${\displaystyle S(G,A)}$ ${\displaystyle G=\mathbb {Z} }$ ${\displaystyle S(G,A)}$ ${\displaystyle A}$

literature

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