Tag Archives: projective modules

Projective Principal Ideals, Idempotent Annihilators

Given an idempotent $e$ in a ring $R$, the right ideal $eR$ is projective as a right $R$-module. In fact, $eR + (1-e)R$ is actually a direct sum decomposition of $R$ as a right $R$-module. An easy nontrivial example is $\Z\oplus\Z$ with $e = (1,0)$. Fix an $a\in R$. If $aR$ is a projective right […]

Examples: Projective Modules that are Not Free

Here are nine examples of projective modules that are not free, some of which are finitely generated. Direct Products Consider the ring $R= \Z/2\times\Z/2$ and the submodule $\Z/2\times \{0\}$. It is by construction a direct summand of $R$ but certainly not free. And it's finitely generated! Another example is the submodule $\Z/2\subset \Z/6$, though this […]

Projectives and the Devious Determinant

The determinant is certainly a fascinating beast. But what is the determinant? Is it a really just a number or a function on matrices? In this post I hope to convince you that the answer is 'no'. In fact, we will see that the determinant, suitably modified, can be used to classify certain types of […]