Calculations in exceptional groups over rings

In the present paper we discuss a major project, whose goal is to develop theoretical background and working algorithms for calculations in exceptional Chevalley groups over commutative rings. We recall some basic facts concerning calculations in groups over fields, and indicate complications arising in the ring case. Elementary calculations as such are no longer conclusive. We describe basics of calculating with elements of exceptional groups in their minimal representations, which allow to reduce calculations in the group itself to calculations in its subgroups of smaller rank. For all practical matters such calculations are much more efficient, than localisation methods. Bibl. – 147 titles.