Let \(A\) be the ring of quantum-Laurent polynomials in \(n\) variables and \(\Lambda\) the subring in which only the first \(r\) variables have inverses. The author’s aim is to describe the lattice of submodules of cyclic \(\Lambda\)-modules. If \(B\) denotes the subring of \(A\) generated by the first \(n-1\) variables, then \(A\) has the form of a skew-Laurent polynomial ring \(B[X_n,X_n^{-1};\alpha_n]\). He proves that any \(A\)-module \(M\) which is finitely generated as \(B\)-module is a projective \(B\)-module. Any finitely generated torsion-free \(A\)-module \(M\) can by change of variables be regarded as a projective \(B\)-module. Let \(H\) be the field of fractions of \(B\) and \(R=H[X_n ,X_n^{-1};\alpha_n]\). Then the embedding \(M\to R\otimes_AM\) (\(M\) as before) induces a natural isomorphism between the submodule lattices \(L_A(M)\leftrightarrow L_R(R\otimes_AM)\). It follows that \(_AM\) is irreducible if and only if \(_R(R\otimes M)\) is irreducible. After some examples the author examines irreducible modules over polynomials and the extensions to the skew fields of fractions. By studying the endomorphism rings of projective ideals, he is able to prove the following theorem. Let \(\Lambda\), \(\Lambda'\) be rings of general quantum polynomials (each in at least 2 variables of which at least one is invertible); if \(\Lambda\), \(\Lambda'\) are Morita equivalent, then they are isomorphic; moreover \(\text{Pic}(\Lambda)\) is trivial. A final result concerns Zariski’s conjecture. This states that if \(R\) is a subring of \(S=k[X_1,\dots, X_n]\) (\(k\) is a field) such that \(S=R[y]\) for some \(y\in S\), then \(R=k[X_1,\dots,X_{n-1}]\). This is still open, but the author proves that for the ring of quantum polynomials it has a positive answer.