We obtain that every saturated proper ideal of the Steinberg algebra \(A_S({\mathcal {G}})\) of an ample groupoid \({\mathcal {G}}\) over an additively idempotent semifield S is an intersection of annihilators of minimal induced semimodules \(\text {Ind}_u(_{S[{\mathcal {G}}^u_u]}{\mathbb {B}})\) and show that each primitive ideal of \(A_S({\mathcal {G}})\) is the annihilator of a minimal induced semimodule \(\text {Ind}_u(_{S[{\mathcal {G}}^u_u]}{\mathbb {B}})\) , where \(S[{\mathcal {G}}^u_u]\) is the group semiring of the isotropy group of \({\mathcal {G}}\) at u over S, \({\mathbb {B}}\) is the Boolean semifield and \(_{S[{\mathcal {G}}^u_u]}{\mathbb {B}}\) is the trivial left \(S[{\mathcal {G}}^u_u]\) -semimodule \({\mathbb {B}}\) . Consequently, Exel’s Effros–Hahn conjecture holds for Steinberg algebras of ample groupoids over additively idempotent semifields.
