Prime \'etale groupoid algebras with applications to inverse semigroup and Leavitt path algebras

In this paper we give some sufficient and some necessary conditions for an 'etale groupoid algebra to be a prime ring. As an application we recover the known primeness results for inverse semigroup algebras and Leavitt path algebras. It turns out that primeness of the algebra is connected with the dynamical property of topological transitivity of the groupoid. We obtain analogous results for semiprimeness.