# 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 RR-module MM is a subset SS of MM such that every element mm of MM has the form

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

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

Examples

  • Mn(R)M_{n}(R) is free in RModR\catMod, with base {eij:1i,jn}\{ e_{ij} : 1\leq i , j \leq n \} (which has n2n^{2} elements).
  • The ring of triangular matrices over RR is free in RModR\catMod, with base {eij:1ijn}\{ e_{ij} : 1 \leq i \leq j \leq n \} (which has n(n+1)/2n(n+1)/2 elements).
  • R[λ]R[\lambda] is a free RR-module, with base {λi:iN}\{ \lambda^{i} : i \in \NN \}.