t-Extending Krasner Hypermodules

Let M be a hypermodule over a hyperring R such that the intersection of any two subhypermodules of M is a subhypermodule of M . We introduce the concept of a t-essential subhypermodule in M relative to an arbitrary subhypermodule T of M , which is called T -t-essential subhypermodule of M . Our aim in this work is to investigate properties of t-essential subhypermodules. We apply this concept to introduce t-extending hypermodules. Examples are provided to illustrate different concepts.


Introduction
The categories of hypergroups, hypermodules and hyperrings have many important roles in hyperstructures. Some authors got many exiting results about these theories. Reader can see references [1], [6], [7], [8] and [10] to get some basic information about the categories of hypergroups, hyperrings and hypermodules. Also reference [13] can be suitable to get some information about theory of rings and modules.
We recall some definitions and theorems from above references which we need to develop our paper.
In this paper, we use  [5]. A non-empty subset K of a hypergroup (H, •) is called subhypergroup, if for all k ∈ K, we have k • K = K • k = K. A hypergroup (H, •) is called commutative if for all x, y ∈ H, then x • y = y • x. A commutative hypergroup (H, •) is said to be canonical, if there exists a unique 0 ∈ H, such that for all x ∈ H, x • 0 = {x}; for all x ∈ H, there exists a unique x −1 ∈ H, such that 0 ∈ x • x −1 ; if x ∈ y • z, then y ∈ x • z −1 and z ∈ y −1 • x, for all x, y, z ∈ H [5].
The triple (R, ⊎, •) is called a hyperring, if (R, ⊎) is a hypergroup, (R, •) is a semihypergroup and • is a distributive over ⊎ [7]. A non-empty subset I of a hyperring R is called a hyperideal if (I, ⊎) is a subhypergroup of (R, ⊎) and r • x ∪ x • r ⊆ I for all x ∈ I and r ∈ R. A hyperring (R, ⊎, •) is called Krasner, if (R, ⊎) is a canonical hypergroup and (R, •) is a semigroup such that 0 is zero element, i.e. for all x ∈ R, we have x • 0 = 0 = 0 • x [7]. A non-empty subset I of a krasner hyperring (R, ⊎, •) is called a a right hyperideal of R if (I, ⊎) is a canonical subhypergroup of (R, ⊎) and for every a ∈ I and r ∈ R, a • r ∈ I [11].
Let (R, ⊎, •) be a hyperring and (M, +) be a hypergroup. If there exists an external hyperoperation . : M × R −→ P * (M ) such that for all a, b ∈ M and r, s ∈ R we have (a + b).r = (a.r) + (b.r), a.(r ⊎ s) = (a.r) + (a.s) and a.(r • s) = (a.r).s then (M, +, .) is called a right hypermodule over R [3] is the second singular subhypermodule of M . Whenever T = ∩ K≤M K and N ☎ T t M , then we call N a t-essential subhypermodule of M and N ☎ t M shows that N is a t-essential subhypermodule of M . In the light of last remarks and comments, in what follows we start to study on t-essentially in R-hypermodules. It can be seen that for a Krasner R-hypermodule M and K ≤ M , we can construct the quotient Krasner R-hypermodule M/K, endowed with (x + K) ⊕ (y + K) = { t + K | t ∈ x + y } and (x + K) ⊙ r = x.r + K, for all x + K, y + K ∈ M/K and r ∈ R.
Let T be a subhypermodule of an R-hypermodule M . In [9], a subhypermodule N of M is called a Tdirect summand provided there exists a subhypermodule K of M such that M = N +K and N ∩K ⊆ T . If

t-Closed Subhypermodules and t-Extending Krasner Hypermodules
In this section, we define notions of t-closed subhypermodules and t-extending Krasner hypermodules, and we obtain some basic properties of t-extending Krasner hypermodules.
t M , then we call N a t-essential subhypermodule of M and will be denoted by Let M and N be two R-hypermodules. Recall from [9] that the function f : .r for all x, y ∈ M and r ∈ R. Note that in this case, f (0 M ) = 0 N . If a strong homomorphism f is one-to-one and surjective function, it is called a strong isomorphism.
We  (1) N is t-essential subhypermodule of M ; (2) (N + Z 2 (M ))/Z 2 (M ) is an essential subhypermodule of M/Z 2 (M );  Proof. (1) ⇒ (2) There exists a subhypermodule K of M such that N ⊕K is an essential subhypermodule of M . By hypothesis, K ≤ Z 2 (M ) therefore; N + Z 2 (M ) is an essential subhypermodule of M , and since Z 2 (M ) is a closed subhypermodule of M , we conclude that (N + Z 2 (M ))/Z 2 (M ) is an essential subhypermodule of M/Z 2 (M ).
(2) ⇒ (3) This is obvious.    (1) There exists a subhypermodule K such that N is maximal with respect to the property that N ∩ K is Z 2 -torsion; (2) N is t-closed in M ;   Proof. (1) By Lemma 2.4 (1) it suffices to show that if N = Z 2 (M ), then there is a t-essential subhypermodule K of M for which N ∩ K ≤ Z 2 (M ). Then there exists a subhypermodule K of M such that N is maximal with respect to the property that N ∩ K is Z 2 -torsion by Proposition 2.5. Let K ∩ T ≤ Z 2 (M ). By Zorn's Lemma, T can be enlarged into a t-closed subhypermodule N ′ such that K ∩ N ′ ≤ Z 2 (M ). In addition, by Lemma 2.4 (1),

Proof. (1) =⇒ (2) Assume that the first condition is provided and
. Then T ≤ Z 2 (M ) and so K is t-essential.

We have in general
for an R-hypermodule M . But above conditions are always true if we replace ≤ c by ≤ tc .
Proposition 2.8. Let M be a Krasner R-hypermodule. Then:

5
Moreover, an arbitrary intersection of t-closed subhypermodules is t-closed.
Let N i be a t-closed subhypermodule of M for any i in an index set I. It is clear that there is a strong monomorphism from M/ ∩ i N i to i M/N i . Then M/N i is nonsingular by Proposition 2.5 (6).
Motivated by the definition of an extending hypermodule, we define the following notion.
It is clear that every Z 2 -torsion hypermodule is t-extending by Lemma 2.4. Moreover, every extending hypermodule is t-extending since every t-closed subhypermodule is closed by Proposition 2.5. The following Theorem gives several equivalent conditions for an hypermodule to be t-extending.
(4) =⇒ (5) Suppose that K is a subhypermodule of M . There exists a direct summand N of M such that K + Z 2 (M ) is essential in N . By Proposition 2.2, K ☎ t N . (1) Every strong homomorphic image of M is t-extending. In particular, every direct summand of M is t-extending.
(2) Every fully invariant subhypermodule of M is t-extending.
Proof. (1) Let K be a subhypermodule of M and L/K ≤ M/K. Since M is t-extending, there exists a direct summand N of M such that L ☎ t N . Therefore, N/K is a direct summand of M/K such that L/K ☎ t N/K by Proposition 2.2 (4). Thus M/K is t-extending by Theorem 2.10 (5).
(2) Let L be a fully invariant subhypermodule of M and K be a subhypermodule of L. There exists a decomposition M = N ⊕ N ′ such that K ☎ t N . Since L is fully invariant, L = (N ∩ L) ⊕ (N ′ ∩ L). However, K ☎ t N ∩ L since (N ∩ L)/K ≤ N/K is Z 2 -torsion. Thus L is t-extending. ✷ The following example shows that the notions of extending and t-extending are not the same. Moreover, this example shows that a t-extending hypermodule not be Z 2 -torsion.
Example 2.13. For an arbitrary Z-hypermodule M , since Z 2 ⊕ Z 8 is not extending, the Z-hypermodule E(M ) ⊕ Z 2 ⊕ Z 8 is t-extending but not extending.