example of a projective module which is not free

Let $R_{1}$ and $R_{2}$ be two nontrivial, unital rings and let $R=R_{1}\oplus R_{2}$ . Furthermore let $\pi_{i}:R\to R_{i}$ be a projection for $i=1,2$ . Note that in this case both $R_{1}$ and $R_{2}$ are (left) modules over $R$ via

$\cdot:R\times R_{i}\to R_{i};$

$(r,s)\cdot x=\pi_{i}(r,s)x,$

where on the right side we have the multiplication in a ring $R_{i}$ .

Proposition. Both $R_{1}$ and $R_{2}$ are projective $R$ -modules, but neither $R_{1}$ nor $R_{2}$ is free.

Proof. Obviously $R_{1}\oplus R_{2}$ is isomorphic (as a $R$ -modules) with $R$ thus both $R_{1}$ and $R_{2}$ are projective as a direct summands of a free module.

Assume now that $R_{1}$ is free, i.e. there exists $\mathcal{B}=\{e_{i}\}_{i\in I}\subseteq R_{1}$ which is a basis. Take any $i_{0}\in I$ . Both $R_{1}$ and $R_{2}$ are nontrivial and thus $1\neq 0$ in both $R_{1}$ and $R_{2}$ . Therefore $(1,0)\neq(1,1)$ in $R$ , but

$(1,1)\cdot e_{i_{0}}=\pi_{1}(1,1)e_{i_{0}}=1e_{i_{0}}=\pi_{1}(1,0)e_{i_{0}}=(1% ,0)\cdot e_{i_{0}}.$

This situation is impossible in free modules (linear combination is uniquely determined by scalars). Contradiction. Analogously we prove that $R_{2}$ is not free. $\square$

Title	example of a projective module which is not free
Canonical name	ExampleOfAProjectiveModuleWhichIsNotFree
Date of creation	2013-03-22 18:49:55
Last modified on	2013-03-22 18:49:55
Owner	joking (16130)
Last modified by	joking (16130)
Numerical id	6
Author	joking (16130)
Entry type	Example
Classification	msc 16D40