  We extend the notion of monogenic extension to the noncommutative setting,
and we study the Hochschild cohomology ring of such an extension. As an
aplication we complete the computation of the cohomology ring of the rank one
Hopf algebras beggined in [S. M. Burciu and S. J. Witherspoon, Hochschild
cohomology of smash products and rank one Hopf algebras].
