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.