On Strong Dual Rickart Modules

Gangyong Lee, S. Tariq Rizvi, and Cosmin S. Roman studied Dual Rickart modules. The main purpose of this paper is to define strong dual Rickart module. Let M and N be Rmodules , M is called Nstrong dual Rickart module (or relatively sd-Rickart to N ) which is denoted by M it is N-sdRickart if for every submodule A of M and every homomorphism f Hom (M , N) , f (A) is a direct summand of N. We prove that for an Rmodule M , if R is M-sdRickart , then every cyclic submodule of M is a direct summand . In particular, if M is projective, then M is Z-regular. We give various characterizations and basic properties of this type of modules.


Introduction.
A module M is called dual Rickart module if for every φ End (M) , then Imφ = eM , for some = e. Equivalently a module M is dual Rickart module if and only if for every φ End (M) , then Imφ is a direct summand of M , See [1] , [2] . Some generalizations of dual Rickart modules and related concepts are recently introduced in [8], [9] and [10]. A module M is Nd-Rickart (or relatively d-Rickart to N) if for every homomorphism φ: M N , Imφ is a direct summand of N , where N is any R-module , see [1]. In this paper , we define strong dual Rickart modules, A module M is called N-strong dual Rickart module (or relatively sd-Rickart to N ) which is denoted by M it is N-sd-Rickart if for every submodule A of M and every homomorphism f Hom (M , N) , f (A) is a direct summand of N. We also give some results on this type of modules.
In section two, we give properties for the relatively strong dual Rickart modules. For example, let M and N be R-modules and let A be a submodule of M.
In section three, we give various characterizations of relatively strong dual Rickart module, we show that if N and M modules, then M is N-sd-Rickart if and only if every short exact sequence Splits , for every submodule A of M and f Hom(M,N) , where i is the inclusion map and is the natural epimorphism.

Strong dual Rickart.
In this section, we investigate and study the notion of relatively strong dual Rickart modules, and we obtain some of fundamental properties, several relations between sd-Rickart modules, and other classes of modules are obtained in this section. (2) Every sd-Rickart module is d-Rickart , the converse is not true in general , for example, Q as Z-module is d-Rickart, by [1] which is not sd-Rickart, because it is not semisemple. (4). The converse is not true in general for example, Z 6 is not relatively -sd-Rickart to Z 12 as Z-module, hence sd-Rickart is not symmetric property. (6) Let M and N be R-modules with Hom(M , N) = 0 , then M is N-sd-Rickart. For example Hom(Q , Z) = 0 implies Q is Z-sd-Rickart. Also , Hom(Z n , Z) = 0 implies Z n is Z-sd-Rickart. (7) One can easily show that 0 is M-sd-Rickart and M is 0-sd-Rickart , for every R-module M.
Recall that an R-module M is said to be coquasi-Dedekind if for every proper submodule A of M , Hom(M,A) =0. Equivalently, M is to be coquasi-Dedekind if every nonzero endomorphism of M is an epimorphism [3].
Now, we study the properties of the N-strong Dual Rickart modules.
Consider the following diagram.
Where i 1 , i 2 , i 3 and i 4 are inclusion maps and p 1 , p 2 , p 3 and p 4 are projection maps. Since

Characterizations of strong dual Rickart modules.
In this section, we give various characterizations of strong dual Rickart modules. We also obtain a characterization for an arbitrary direct sum of relatively sd-Rickart modules.
We start this section by the following proposition.  Where i is the inclusion map and p is the projection map. Since X is a submodule of M and M is N-sd-Rickart, then (i is a direct summand of A. Thus , we get the result. Where i is the inclusion map. Clearly that (X) is a direct summand of L.  (1) M is R-sd-Rickart, for every R-module M.