Rigidity and triangularity of an exponential map

Let [Formula: see text] be a field of arbitrary characteristic, [Formula: see text] be a domain, and [Formula: see text]. Then (1) All exponential maps of [Formula: see text] are rigid, and we give a necessary and sufficient condition for the triangularity of [Formula: see text]. (2) If [Formula: see text] such that [Formula: see text], then [Formula: see text] is rigid and we give a necessary and sufficient condition for the triangularity of [Formula: see text]. When [Formula: see text] is of zero characteristic, [Formula: see text] is due to [D. Daigle, A necessary and sufficient condition for triangulability of derivations of [Formula: see text], J. Pure Appl. Algebra 113(3) (1996) 297–305] and [Formula: see text] is due to [M. K. Keshari and S. A. Lokhande, A note on rigidity and triangulability of a derivation, J. Commut. Algebra 6(1) (2014) 95–100].