explicit definition of polynomial rings in arbitrarly many variables

Let R be a ring and let 𝕏 be any set (possibly empty). We wish to give an explicit and formal definition of the polynomial ringMathworldPlanetmath R⁒[𝕏].

We start with the set

β„±(𝕏)={f:𝕏→ℕ|f(x)=0Β for almost allΒ x}.

If 𝕏={X1,…,Xn} then the elements of ℱ⁒(𝕏) can be interpreted as monomialsMathworldPlanetmathPlanetmath


Now define

R[𝕏]={F:β„±(𝕏)β†’R|F(f)=0Β for almost allΒ f}.

The additionPlanetmathPlanetmath in R⁒[𝕏] is defined as pointwise addition.

Now we will define multiplication. First note that we have a multiplication on ℱ⁒(𝕏). For any f,g:𝕏→ℕ put


This is the same as multiplying xaβ‹…xb=xa+b.

Now for any fβˆˆβ„±β’(𝕏) define


Now if F,G∈R⁒[𝕏] then we define the multiplication


by putting


Note that all of this well-defined, since both F and G vanish almost everywhere.

It can be shown that R⁒[𝕏] with these operationsMathworldPlanetmath is a ring, even an R-algebraMathworldPlanetmathPlanetmath. This algebra is commutativePlanetmathPlanetmathPlanetmath if and only if R is. Furthermore we have an algebra homomorphism


which is defined as follows: for any r∈R let Fr:ℱ⁒(𝕏)β†’R be the function such that if f:𝕏→ℕ is such that f⁒(x)=0 for any xβˆˆπ•, then put Fr⁒(f)=r and for any other function fβˆˆβ„±β’(𝕏) put Fr⁒(f)=0. Then


is our function, which is a monomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. Furthermore if R is unital with the identityPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath 1, then


is the identity in R⁒[𝕏]. Anyway we can always interpret R as a subset of R⁒[𝕏] if put r=Fr for r∈R.

Note, that if 𝕏=βˆ…, then R⁒[βˆ…] is still defined and E:Rβ†’R⁒[𝕏] is an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of rings (it is ,,onto”). Actually these two conditions are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath.

Also note, that 𝕏 itself can be interpreted as a subset of R⁒[𝕏]. Indeed, for any xβˆˆπ• define


by fx⁒(x)=1 and fx⁒(y)=0 for any yβ‰ x. Then define


by putting Fx⁒(fx)=1 and Fx⁒(f)=0 for any fβ‰ fx. It can be easily seen that Fx=Fy if and only if x=y. Thus we will use convention x=Fx.

With these notations (i.e. R,π•βŠ†R⁒[𝕏]) we have that elements of R⁒[𝕏] are exactly polynomialsMathworldPlanetmathPlanetmath in the set of variables 𝕏 with coefficients in R.

Title explicit definition of polynomial rings in arbitrarly many variables
Canonical name ExplicitDefinitionOfPolynomialRingsInArbitrarlyManyVariables
Date of creation 2013-03-22 19:18:10
Last modified on 2013-03-22 19:18:10
Owner joking (16130)
Last modified by joking (16130)
Numerical id 8
Author joking (16130)
Entry type Definition
Classification msc 12E05
Classification msc 13P05
Classification msc 11C08