  In the arithmetic of function fields Drinfeld modules play the role that
elliptic curves take on in the arithmetic of number fields. As higher
dimensional generalizations of Drinfeld modules, and as the appropriate
analogues of abelian varieties, G. Anderson introduced pure t-motives. In this
article we study the arithmetic of the later. We investigate which pure
t-motives are semisimple, that is, isogenous to direct sums of simple ones. We
give examples for pure t-motives which are not semisimple. Over finite fields
the semisimplicity is equivalent to the semisimplicity of the endomorphism
algebra, but also this fails over infinite fields. Still over finite fields we
study the endomorphism rings of pure t-motives and criteria for the existence
of isogenies. We obtain answers which are similar to Tate's famous results for
abelian varieties.
