free modules over a ring which is not a PID

Let R be a unital ring. In the following modules will be left modules.

We will say that R has the free submodule property if for any free modulePlanetmathPlanetmath F over R and any submodule FF we have that F is also free. It is well known, that if R is a PID, then R has the free submodule property. One can ask whether the converseMathworldPlanetmath is also true? We will try to answer this question.

PropositionPlanetmathPlanetmath. If R is a commutative ring, which is not a PID, then R does not have the free submodule property.

Proof. Assume that R is not a PID. Then there are two possibilities: either R is not a domain or there is an ideal IR which is not principal. Assume that R is not a domain and let a,bR be two zero divisors, i.e. a0, b0 and ab=0. Let (b)R be an ideal generated by b. Then obviously (b) is a submodule of R (regarded as a R-module). Assume that (b) is free. In particular there exists m(b), m0 such that rm=0 if and only if r=0. But m is of the form λb and because R is commutativePlanetmathPlanetmathPlanetmathPlanetmath we have


ContradictionMathworldPlanetmathPlanetmath, because a0. Thus (b) is not free although (b) is a submodule of a free module R.

Assume now that there is an ideal IR which is not principal and assume that I is free as a R-module. Since I is not principal, then there exist a,bI such that {a,b} is linearly independentMathworldPlanetmath. On the other hand a,bR and 1 is a free generatorPlanetmathPlanetmath of R. Thus {1,a} is linearly dependent, so


for some nonzero λ,αR (note that in this case both λ,α are nonzero, more precisely λ=a and α=-1). Multiply the equation by b. Thus we have


Note that here we used commutativity of R. Since {a,b} is linearly independend (in I), then the last equation implies that λ=0. Contradiction.

Corollary. Commutative ring is a PID if and only if it has the free submodule property.

Title free modules over a ring which is not a PID
Canonical name FreeModulesOverARingWhichIsNotAPID
Date of creation 2013-03-22 18:50:08
Last modified on 2013-03-22 18:50:08
Owner joking (16130)
Last modified by joking (16130)
Numerical id 5
Author joking (16130)
Entry type Definition
Classification msc 13E15