T-Polyform Modules

We introduce the notion of t-polyform modules. The class of tpolyform modules contains the class of polyform modules and contains the class of t-essential quasi-Dedekind. Many characterizations of t-polyform modules are given. Also many connections between these class of modules and other types of modules are introduced.


Introduction
Throughout the paper, rings will have a nonzero identity element and modules will be unitary right modules. We first briefly review some background materials relevant to the topics discussed in this paper.
Recall that, a submodule N of an R-module M is called essential submodule of M( briefly Obviously, every essential submodule is t-essential, but not conversely, for example the submodule ) 4 ( of the z-module is t-essential but not essential. However, the two concepts are equivalent if M is nonsingular (ie Z(M)=0). A module M is called singular if Z(M)= M and is called Z 2 -torslon if . If A M then Asgari..etc, in [2], introduced the concept t-closed submodule where a submodule N of an R-Module M is t-closed (denoted by M N tc  if N has no proper t-essential extension in M. It is clear that every t -closed submodule is closed, but the converse is not true for example ( 0 ) is closed in Z 8 as Zmodule but it is not t-closed. The two concepts closed submodule and t-closed submodule are coincide in nonsignular modules. It is clear that polyform module implies K-nonsingular but not conversely see [5]. Thaa'r in [4] gave the notion of essentially quasi-Dedekind modules as a generalization of quasi -Dedekind modules by restricting the definition of quasi-Dedekind modules (which is introduced in [ [7]proved that k-nonsingular modules and essentially quasi-Dedekind are coincided. F,S and Inaam in [8]  It is obvious that every t-ess. q.Ded module is ess. q-Ded, but not conversely [8,Rem&Ex.2.2(2)]. In the present paper, motivated by these works, we introduce and study t-polyform modules as follows: An R-module M is called t-polyform if for each L M, and :L→M, Next note that our notion ((t-polyform modules)) is different from (st-polyform modules) which is appeared recently in [9] as we explain that in S.3, Note 3.5

2-Preliminaries
We list some known results which are relevant for our work.

Lemma 2.1 [2]
The following statements are equivalent for a submodule A of an R-module M. 1.

Lemma 2.6 [2]
Let . A ring R is said to be right t-polyform if the module R R is t-polyform.

3-t-polyform Modules
Remarks and Examples 3.2 1.Every t-polyform module is polyform, since every essential submodule is t-essential. However, the converse is not always true for example: Let M be the Z-module Z 6 since M has no proper essential submodule, then M is polyform. But M is singular hence M is Z 2 -torsion and so every submodule, 0 L M is Z 2 -torsion and hence Thus M=Z 6 is not t-polyform 2. It is known that every semisimple module is polyform,but it is not necessary t-polyform, see the example in (1) .
In particular each of the Z-module: Z,Q,Z Z,Q , Z[X] is t-polyform module, also for each prime number P, Z p as Z p -module is t-polyform.   [11,12].

10.Recall that an R-module is
If M is t-polyform and Co-epi-retractable, then N M is t-polyform, for each N .

Proof: it follows directly
The following theorem is a characterization of t-polyform modules.  The following is another characterization of t-polyform modules The notion of ((st-polyform modules)) appeared in [9], where an R-module M is called st-polyform if for each 0   [1]. Note that every rational submodule is essential but not conversely [1] We give the following: Theorem 3.6 An R-module M is t-polyform implies every nonzero t-essential submodule of M is rational.

Remark 3.7
The converse of theorem (3.6) is not true in general, for example: The Z-module Z 6 is not t-polyform, but Z 6 has only Z 6 as t-essential submodule of Z 6 and Recall a nonzero R-module M is called monoform if for each 0 and for each 0 (N,M),then ker f=0 , [9]. Equivalently a nonzero R-module M is monoform if for each nonzero submodule N of M, M N r  , [9].
It is known that every monoform is polyform . Now we ask the following: Is there any relation between t-polyform modules and monoform?
Consider the following remarks Remarks 3.9 1. t-polyform modules need not be monoform, for example: The Z-module is t-polyform (Rem 3.2.(4)), but it is not monoform since there exists such that f(x,y)=(y,0) for each xZ, y2Z then Kerf= zero submodule. 2. Monoform module may be not t-polyform module, for example: The Z-module Z p , where p is a prime number, is monoform but it is not t-polyform.
We introduce the following Definition3.10 An R-module M is called t-essentialy monoform (shortly t-ess-mono) if for each 0