Definition of the Quantum Plane

Question

When ever I find definition of the quantum plane it says $A_q^2 = C\langle x,y \rangle/I$, where $I = C\langle xy-qyx \rangle$. What I want to know is whether they mean the unital free algebra or just the free algebra. In brief, is the quantum plane unital?

Moreover, when people write $A_q^N$, they mean the free (unital) algebra with $N$ generaterators, where every generator just commutes with every other generator?

Answer 1

Kassel, p. 3 ($k$ is the ground field):

An algebra is a ring together with a ring map $\eta_A : k \to A$ whose image is contained in the center of $A$. ... A morphism of algebras or an algebra morphism_ is a ring map $f : A \to B$ such that $f \circ \eta_A = \eta_B.$ (1.1) As a consequence of (1.1), $f$ preserves the units, i.e., we have $f(1) = 1$.

There is also a definition of free algebras on p. 7 where he explicitly states that they have a unit.