Decomposable right half smash product spaces

It is shown that for any space A, the cofibration X \to X \Join \sumA \to \sumA \wedge X$ decomposable when X is a co-T-space. It is also obtain necessary and sufficient conditions for the cofibration $X \to X \Join A \to A \wedge X$ is trivial, in the sense of cofibre homotopy type.