The coupled q-oscillator algebra \({\cal A}\) is a smash product of the polynomial algebra in one variable with the Hopf algebra \(U_{q}({\mathfrak s}{\mathfrak l}_{2})\) . We show that the centre of the algebra \({\cal A}\) is trivial. We find a distinguished normal element of \({\cal A}\) that plays a role in studying its structures and representations. We give explicit descriptions of the prime, completely prime, primitive and maximal ideals of the algebra \({\cal A}\) . As a result, we show that \({\cal A}\) cannot have a Hopf algebra structure, and \({\cal A}\) has no finite-dimensional representations. The group of automorphisms of \({\cal A}\) is explicitly described. A classification of all simple weight modules over the algebra \({\cal A}\) is obtained. Using the classification of primitive ideals of \({\cal A}\) , we determine the annihilator of every simple weight module.
