I am studying the non-commutative torus $ A\_{\theta} $.

When $ \theta $ is irrational, $ {K\_{0}}(A\_{\theta}) $ is generated by $ [1] $ and $ [p\_{\theta}] $.

(**Note:** $ p\_{\theta} $ is a projection in $ A\_{\theta} $, called the *Powers-Rieffel projection*, and satisfies $ \tau(p\_{\theta}) = \theta $, where $ \tau $ is the unique tracial state on $ A\_{\theta} $.)

When $ \theta $ is rational, we have the same $ K $-theory as $ A\_{\theta} $ is Morita equivalent to $ C(\mathbb{T}^{2}) $. However, I would like to know an actual generating set for $ {K\_{0}}(A\_{\theta}) $. Could anyone give an example or provide a reference?

I thank you all in advance.