  We investigate the representation theory of the Temperley-Lieb algebra,
$TL_n(\delta)$, defined over a field of positive characteristic. The principle
question we seek to answer is the multiplicity of simple modules in cell
modules for $TL_n$ over arbitrary rings. This provides us with the
decomposition numbers for this algebra, as well as the dimensions of all simple
modules. We obtain these results from diagrammatic principles, without
appealing to realisations of $TL_n$ as endomorphism algebras of
$U_q(\mathfrak{sl}_2)$ modules. Our results strictly generalise the known
characteristic zero theory of the Temperley-Lieb algebras.
