Prime étale groupoid algebras with applications to inverse semigroup and Leavitt path algebras. (English) Zbl 1471.20044
Summary: In this paper we give some sufficient and some necessary conditions for an étale 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.
MSC:
| 20M18 | Inverse semigroups |
| 20M25 | Semigroup rings, multiplicative semigroups of rings |
| 16S88 | Leavitt path algebras |
| 16S36 | Ordinary and skew polynomial rings and semigroup rings |
| 22A22 | Topological groupoids (including differentiable and Lie groupoids) |
| 18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |
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