We introduces the concept of dual Rickart (Baer) modules in relation to the cosingular submodule. The paper demonstrates that a module is considered to be Z-dual Rickart only if its submodule, Z(M), is a dual Rickart direct summand of the module M. Additionally, it is proven that a module is considered dual Baer with respect to Z(M) only when it is dual Rickart with respect to Z(M), and the module has the strong summand sum property for direct summands included in Z(M). Lastly, we present a characterization of right Z-dual Baer rings.
