Engel varieties of associative rings and the number of Mersenne primes

An associative ring R can be viewed as a semigroup via a b=a+b+ab, and as a Lie ring via [a,b]=ab−ba. It is known that the Lie ring [R] is nilpotent if and only if the adjoint semigroup (R, ) is nilpotent (in the sense defined by Malʹcev or by Neumann and Taylor). We prove a similar result for associative rings whose Lie rings satisfy an Engel identity. Mersenne primes appear in an unexpected role.