free objects in the category of commutative algebras

Let R be a commutative ring and let ๐’œโขโ„’โข๐’ขcโข(R) be the categoryMathworldPlanetmath of all commutative algebras over R and algebra homomorphisms. This category together with the forgetful functorMathworldPlanetmathPlanetmath is a construct (i.e. it is a concrete category over the category of sets ๐’ฎโขโ„ฐโข๐’ฏ). Therefore we can talk about free objects in ๐’œโขโ„’โข๐’ขcโข(R) (see this entry ( for definitions).

Theorem. For any set ๐• the polynomial algebra Rโข[๐•] (see parent object) is a free object in ๐’œโขโ„’โข๐’ขcโข(R) with ๐• being a basis. This means that for any commutative algebra A and any function


there exists a unique algebra homomorphism F:Rโข[๐•]โ†’A such that


for any xโˆˆ๐•.

Proof. Assume that f:๐•โ†’A is a function. If WโˆˆRโข[๐•], then there are finite subsets A1,โ€ฆ,AnโŠ†๐• (not necessarily disjoint) and natural numbersMathworldPlanetmath nโข(x,i), i=1,โ€ฆ,n such that W can be uniquely expressed as


with ฮปiโˆˆR. Define Fโข(W) by putting


Of course F is well defined and obviously Fโข(x)=fโข(x). We leave as a simple exercise that F is an algebra homomorphism. The uniqueness of F again follows from the explicit form of W. It is easily seen that Fโข(W) depends only on Fโข(x) for xโˆˆ๐•. This completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof. โ–ก

Corollary 1. If ๐• is a set and ๐•โŠ†๐•, then the inclusion i:๐•โ†’๐• induces an algebraMathworldPlanetmathPlanetmath monomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath


In particular we can treat Rโข[๐•] as a subalgebraMathworldPlanetmathPlanetmath of Rโข[๐•].

Proof. We have a well-defined function i:๐•โ†’Rโข[๐•], iโข(y)=y. By the theorem we have an extensionPlanetmathPlanetmathPlanetmathPlanetmath


such that Iโข(y)=y. It remains to show, that I is ,,1-1โ€. Indeed, assume that Iโข(W)=0 for some polynomialMathworldPlanetmathPlanetmath WโˆˆRโข[๐•]. But if we recall the expression of W as in proof of the theorem and remember that I is an algebra homomorphism, then it is easy to see that Iโข(y)=y implies that


In particular W=0, which completes the proof. โ–ก

Corollary 2. If A is an R-algebra, then there exists a set ๐• such that


for some ideal I.

Proof. Let ๐•=A as a set. Define


by fโข(x)=x. By the theorem we have an algebra homomorphism


such that Fโข(x)=x for xโˆˆ๐•. In particular F is ,,ontoโ€ and thus by the First Isomorphism TheoremPlanetmathPlanetmath for algebras we have


which completes the proof. โ–ก

Title free objects in the category of commutative algebras
Canonical name FreeObjectsInTheCategoryOfCommutativeAlgebras
Date of creation 2013-03-22 19:18:13
Last modified on 2013-03-22 19:18:13
Owner joking (16130)
Last modified by joking (16130)
Numerical id 6
Author joking (16130)
Entry type Theorem
Classification msc 13P05
Classification msc 11C08
Classification msc 12E05