STRONGLY T-EXTENDING MODULES AND STRONGLY T-BAER MODULES

In this paper, we introduce the notions of strongly t-extending and strongly t-Baer modules. We provide several characterizations and investigate properties of each of these concepts. It is shown that, while a direct summand of a strongly t-extending module inherits the property, direct sums of strongly t-extending modules do not. Moreover, when a direct sum of strongly t-extending modules is strongly t-extending, is investigated. Also, it is proved that every strongly t-extending module has strongly summand intersection property and densely co-Hopfian property.


Introduction
The idea of investigating a mathematical structure via its representations in simpler structures is commonly used and often successful. The representation theory of extending modules and Baer modules has developed greatly in the recent years.
It is an area which is very firmly based on the detailed understanding of examples, and there are many powerful techniques for investigating representations of particular extending modules (resp. Baer modules) and for relating representations of different extending modules (resp. Baer modules) to one another. One point of this paper is to introduce a subclass of t-extending modules (resp. t-Baer modules).
The notion of an extending module can be traced back to work of von Neumann in the 1930s. His interest in quantum mechanics led him to develop "continuous geometry", which we today refer to as upper and lower continuous complete modular lattices. In recent years theory of extending modules and rings has come to play an important role in the theory of rings and modules. A module M is called extending if every submodule is essential in a direct summand. Many properties of extending modules have been introduced and studied by several authors. The Baer property for rings was first considered by Kaplansky [11]. A ring R is called Baer if the right annihilator of any subset of R is generated as a right ideal by an idempotent. The notion of Baer property in a general module theoretic setting has been introduced by Rizvi and Roman. In [13], a strong connection is established between extending modules and Baer modules.
In [3], Asgari and Haghany introduced the concept of t-extending modules and t-Baer modules by using second singular submodules. The notion of a strongly extending module is introduced in [9], which is a subclass of the class of extending modules. Motivated by definitions of strongly extending modules and t-extending modules, we introduce the notion of strongly t-extending modules which are particular t-extending modules and a generalization of strongly extending modules.
After some preliminaries in Section 2, we propose the definition of a strongly t-extending module in Section 3. We explore some equivalent conditions for a module to be strongly t-extending. It is shown that direct summands of strongly t-extending modules are too strongly t-extending. A natural question to ask, for strongly t-extending modules in whether the property is preserved by a direct sum of such modules. First we answer this in the negative by an example. Next we give a necessary condition for a sum of strongly t-extending modules to be strongly t-extending. Also it is shown that strongly t-extending modules are densely co-Hopfian.
We define and investigate strongly t-Baer modules in Section 4, which were motivated by definitions of t-Baer modules and Abelian Baer modules. We give characterizations of a strongly t-Baer notion. We show that every direct summand of a strongly t-Baer module is strongly t-Baer. Necessary and sufficient conditions are given to show that a direct sum of strongly t-Baer modules is strongly t-Baer.

Preliminaries
Throughout all rings (not necessarily commutative rings) have identities and all modules are unital right modules. For the sake of completeness, we state some definitions and notations used throughout this paper. Let M be a module over a ring R. For submodules N and K of M , N ≤ K denotes N is a submodule of K and End(M ) denotes the ring of right R-module endomorphisms of M . We denote by r M (.) the right annihilator of a subset of End(M ) with elements from M . In what follows, by ≤ ⊕ and ≤ ess we denote, respectively, a module direct summand and an essential submodule of M . The symbols Z, Z n and Q stand for the ring of integers, the ring of residues modulo n and the ring of rational numbers, respectively.
Recall that the singular submodule Z(M ) of a module M is the set of m ∈ M with r R (m) ≤ ess R R , or equivalently, mI = 0 for some essential right ideal I of R. The second singular (or Goldie torsion) submodule Z 2 (M ) is the submodule (d) A submodule C is called t-closed (resp. closed ) if C has no t-essential (resp. essential) extension in M . The symbol C ≤ tc M denotes that C is a t-closed submodule of M ( [3]).
(e) A module M is called t-extending if every t-closed submodule of M is a direct summand of M ( [3]).
is a direct summand of M for each left ideal I of End(M ) ( [3]).
(g) An R-module M is said to be strongly t-Rickart, if t M (φ) is a fully invariant direct summand of M for each φ ∈ End(M ). An R-module M is said to be strongly Rickart, if Ker(φ) is a fully invariant direct summand of M for each φ ∈ End(M ) ( [8,9]).  (k) An idempotent e ∈ R is called left semicentral if re = ere, for each r ∈ R, equivalently, eR is an ideal of R. The set of all semicentral idempotents of R will be denoted by S l (R). If e 2 = e ∈ End(M ), then e ∈ S l (End(M )) if and only if eM is a fully invariant direct summand ( [6,7]).
It is known from [2, Proposition 1.1] that We need the following propositions proved in [9,Theorem 2.4]  (b) The following statements are equivalent for a submodule N of M .
The following statements are equivalent:

Strongly t-extending modules
We start this section with the following definition. Clearly, every Z 2 -torsion module is strongly t-extending. Moreover, every strongly extending module is strongly t-extending. The next result gives several equivalent conditions for a module to be strongly t-extending.
Theorem 3.2. The following are equivalent for an R-module M .
(1) M is strongly t-extending; (2) Every t-closed submodule of M is a fully invariant direct summand; (3) M is t-extending and each direct summand of M which contains Z 2 (M ) is fully invariant; where M is a strongly extending module; (5) Every submodule of M which contains Z 2 (M ) is essential in a fully invariant direct summand; (2) ⇒ (3) Since every t-closed submodule of M is a direct summand, M is textending. We will prove that each direct summand of M which contains Z 2 (M ) is , K is t-closed in M and so by (2), K is fully invariant.
where M is extending (by [3,Theorem 2.11]). We will prove that M is strongly extending. By Proposition 2.2(a), it suffices to show that each direct summand of M is fully invariant in M .    (7), there exists a fully invariant direct The next example shows that strongly t-extending modules need not be strongly extending. By Theorem 3.2, each strongly t-extending module is t-extending, but the converse does not hold in general, as the following example shows. Then Z 2 (M ) ⊕ R is a t-extending module which is not strongly t-extending since R R is not strongly extending.
Theorem 3.5. If M is a strongly t-extending module, then each direct summand of M is strongly t-extending.
Proof. Let N ≤ ⊕ M , say M = N ⊕ K. Since M is strongly t-extending, M is t-extending, so are N and K by [3,Proposition 2.14]. We will show that each direct  (3), H ⊕ Z 2 (K) is a fully invariant direct summand of M . We will prove that H is fully invariant in N . If f ∈ End(N ), then 1 K ⊕ f ∈ End(M ).
Thus ( Thus N is t-extending and each direct summand of N that contains Z 2 (N ) is fully invariant, therefore N is strongly t-extending, by Theorem 3.2(3).
In [9], it is shown that strongly extending modules are wcH. The following proposition shows that strongly t-extending modules are dcH.
Theorem 3.6. If M is strongly t-extending, then M is dcH.
and f (m)I = 0 for some I ≤ tess R R . Since f is a monomorphism, mI = 0, and hence m ∈ Z 2 (M ). Thusf is a monomorphism. Since  Suppose that ∩ i M i = 0 and so ∩ i M i ≤ tess eM for some e ∈ S l (End(M )). Since Proposition 2.2(b). Therefore for each In the following, for a free module F , rank(F ) denotes the minimum cardinality of any basis of F . Theorem 3.8. Let R be a ring and F be a free R-module. The following are equivalent.
In general, a direct sum of strongly t-extending modules need not be a strongly t-extending module, as the following example shows. We can, however, provide some necessary conditions for a direct sum of any two modules to be strongly t-extending.
Proof. The necessity is clear. For the sufficiency, let K be a t-closed submodule in M . By Lemma 2.3, there exists a t-closed submodule L of K such that K ∩ M 1 ≤ tess L. Since L ≤ tc K ≤ tc M , L is t-closed in M by Proposition 2.2(d). As By assumption L is a fully invariant direct summand of M , say M = L ⊕ L . By modular law K = L ⊕ (K ∩ L ). By Lemma 2.3, there exists a t-closed submodule Since N is strongly t-extending and is fully invariant in M . As L is strongly t-extending and Z 2 (L) = Z 2 (M ), L = Z 2 (M ) ⊕ L 1 by Theorem 3.2. Now, we will show that K is fully invariant in M .
The next theorem gives a condition that a direct sum of two strongly t-extending modules is strongly t-extending. Proof. Let K be a t-closed submodule of M such that K ∩ M 1 ⊆ Z 2 (M ). By ).

Strongly t-Baer modules
The purpose of this section is to introduce the concept of strongly t-Baer modules which are particular t-Baer modules, and study some basic properties of this new class of modules. For the rest of the article, M is an R-module and S = End(M R ). Clearly every Z 2 -torsion module is strongly t-Baer. Moreover, the notions of Abelian Baer and strongly t-Baer coincide for every nonsingular module. In particular, every Abelian Baer ring is strongly t-Baer (because every Abelian Baer ring is nonsingular).
The next theorem states some equivalent conditions for a strongly t-Baer module.
Theorem 4.2. The following statements are equivalent for a module M .
Let K be a direct summand of M which contains Z 2 (M ). We will show that K is fully invariant. Let K = eM for some idempotent e ∈ S. Since t M (1 − e) = t M (S(1 − e)) = eM = K and M is strongly t-Baer, K is fully invariant. (1) ⇒ (5) It is clear. By Theorem 4.2(5), every strongly t-extending module is strongly t-Baer. The converse is not true in general.