A new member of Nicholson's morphic folks

The main aim of the paper is to describe the structure of modules and corresponding rings satisfying the property (P), which says that $M/\mathrm{im}\left(\alpha \right)$ is embeddable into $\ker \left(\alpha \right)$ for each endomorphism α and which generalizes the morphic property. In particular, it is proved that the class of rings with the property (P) is closed under taking products and summands and contains unit regular rings. We also explain connections between the virtually internal cancellation property and the property (P) and characterize the structure of particular classes of rings satisfying the property (P).