# # Constructions of Rings and Modules

## # Free Modules and Rings

Intuitively, "free" objects in a category are constructed in the most general way possible, without extra conditions, hence they "lie" above all over objects.

### # Free Modules

To elucidate further on the concept of an object being "free", we first shall consider a module freely generated by a set, since it has no superfluous condition in its construction.

Definitions

A base for an $R$-module $M$ is a subset $S$ of $M$ such that every element $m$ of $M$ has the form

$m = \sum_{s \in S}r_{s}s,$

where almost all of the $r_{s}\in R$ are zero and are uniquely determined. A module with a base is called free.

Examples

• $M_{n}(R)$ is free in $R\catMod$, with base $\{ e_{ij} : 1\leq i , j \leq n \}$ (which has $n^{2}$ elements).
• The ring of triangular matrices over $R$ is free in $R\catMod$, with base $\{ e_{ij} : 1 \leq i \leq j \leq n \}$ (which has $n(n+1)/2$ elements).
• $R[\lambda]$ is a free $R$-module, with base $\{ \lambda^{i} : i \in \NN \}$.
Last Updated: 12/21/2019, 3:04:48 AM