More generally, example of a projective rank 1 module that isn’t unidimensional can be written as A ⊕ R over R, where A is a non-trivial direct summand of R. Note that this type of example requires the ground ring R to be not simple.

A projective rank 1 module is not necessarily unidimensional: a counterexample can be constructed using the fact that ℤ₂ ⊕ ℤ₃ = ℤ₆ (both as rings and as ℤ₆-modules; the statement generally known as the Chinese remainder theorem). ℤ₂ ⊕ ℤ₆ is a projective ℤ₆-module (its complement to a free module ℤ₆² is ℤ₃), and it obviously has the rank 1. But it isn’t unidimensional since has non-trivial projections among ℤ₆-endomorphisms.

Thanks. Could you write proper $\langle$angle brackets$\rangle$ for the bra-ket notation, not ugly substitution characters that cause erroneous spacing in LaTeX? Also, a hyphen spelled after the “bi” prefix is an extremely rare case (or a substandard spelling).