# 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}$

${\ displaystyle R [G]: = \ {\ alpha \ colon G \ to R {\ {\ big |} \} \ alpha (x) = 0 {\ text {for all but a finite number}} x \} \,}$

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

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}$

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

${\ displaystyle R [G]}$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

• 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}$${\ displaystyle \ alpha \ gamma <\ beta \ delta,}$
so be
${\ displaystyle S (G, A) = \ {f \ colon G \ to A \ mid {\ text {supp}} f {\ text {well-ordered}} \}}$
with With the convolution as multiplication and the component-wise addition becomes a ring. If there is a body, then it is an oblique body . If, for example, is with the natural order, then the ring of formal Laurent series with coefficients in .${\ displaystyle \ mathrm {supp} \, f: = \ {g \ in G \ mid f (g) \ not = 0 \}.}$${\ 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 )