 While identity can of course be added to intuitionistic logic, for applications (e.g., to arithmetic) the equality symbol is generally treated as a distinguished predicate constant satisfying the axioms for an equivalence relation (reflexivity, symmetry and transitivity) and additional nonlogical axioms (e.g., the primitive recursive definitions of addition and multiplication). Identity is decidable, intuitionistically as well as classically, but intuitionistic extensional equality is not always decidable; see the discussion of Brouwer’s continuity axioms in Section 3 of the entry on  intuitionism in the philosophy of mathematics.