The Jacobson Radical of the Endomorphism Semiring of P.Q.- Principal Injective Semimodules

In this work, we introduced the Jacobson radical (shortly Rad (Ș)) of the endomorphism semiring Ș = EndR(B), provided that B is principal P.Q.injective semimodule and some related concepts, we studied some properties and added conditions that we needed. The most prominent result is obtained in section three -If B is a principal self-generator semimodule, then Z(ȘȘ) = W(Ș). Subject Classification: 16y60


Introduction:
This paper is interested in the generalization of some results in ring theory and module theory. Semirings and semimodules and their applications; grow in different branches of mathematics, computer sciences, physics, also in many other areas of modern science. The study of semimodules over semiring has been extensively considered, as reviewed by Golan in (1) and references therein. Semirings are moved from rings but simultaneously there are important differences between them. A semiring is a nonempty set ℛ together with two operations, addition and multiplication, where these two operations are associative, whereas addition is a commutative operation, the distribution law holds, there is 0 ∈ ℛ (additive identity element) such that t+0 = t = 0+t, t0 = 0t = 0 for each t in ℛ and there is multiplicative identity element (denoted 1) where 1≠0. It is commutative if the second operation is commutative. For instance, the set of natural number ℕ is a commutative semiring under usual addition and multiplication, but it is not ring. A semimodule ℬ over semiring ℛ is defined similarly in module over ring (1). In 1945 Nathan Jacobson was the first to study the Jacobson radical for arbitrary rings, it is denoted by J(R) or rad(R), so it was called after his name, where he defined the College of Education for Pure Sciences, University of Babylon, Babylon, Iraq. * Corresponding author: khitam sahib25@gmail.com. * ORCID ID: https://orcid.org/5568-0692-0003-0000 Jacobson radical of a ring R to be the ideal consisting of those elements in R that annihilate left R-module and then was given by Kasch(1982) (2), Anderson(1992), Isaacs(1993) and Lam(2001) with equivalent definitions. They defined J(R) to be the intersection of all maximal left ideals of the ring which equals the sum of all superfluous left ideals. Analogously, in this paper we study the Jacobson radical of endomorphism semiring Ș of an ℛsemimodule, in particular of P.Q.-injective semimodule (3). For Ș = ℛ (ℬ), we define the Jacobson radical of Ș, socle semimodule, singular and symbolizes them respectively by Rad(Ș), Soc(ℬ), (Ș). Rad(Ș)= ⋂{ J: J is maximal left ideals of Ș}= the set of all non-invertible elements of Ș (4). (Ș) ={s∈ Ș | ann(s) is essential ideal of Ș}(5), Soc(ℬ)= ∩{Ų | Ų is essential subsemimodule of ℬ}=∑{ ∶ is a simple ℛsubsemimodule of ℬ}(3). Also we define the concept "Kacsh semimodule" similar to what is defined in module (6), as well the concept principally self-generator semimodule and we show that, if ℬ is principally self-generator with Ș = ℛ (ℬ ) we have (Ș ) = W(Ș) where W(Ș) ={ ∈ Ș | ker is essential in ℬ }. We define condition for ℬ called C 2 -condition, where an ℛ-semmimodule ℬ is said to satisfy the C 2 -condition if each subsemimodule of ℬ which is isomorphic to a direct summand of ℬ is itself a direct summand of ℬ, it is shown that every cyclic P.Q.-injective semimodule has C 2 -condition and, if ℬ has C 2 -condition then W(Ș)⊆ Rad(Ș) in module theory, but in semimodule theory we must add some conditions for ℬ in order to get W(Ș)⊆ Rad(Ș) .
Throughout this paper all semirings are commutative with identity and all semimodules are unitary.
In the following, we review some definitions and remarks that will be applied in this paper.

Preliminaries Definition 1. (3).
Let ℛ be a semiring, then for any a ∈ ℛ, ℛa = {x: x =ta} for some t ∈ ℛ. It is a left ideal of ℛ called the principal left ideal generated by a.  . Let ℛ be a semiring and Ų ≤ ℬ, then Ų is said to be a direct summand of ℬ if there exists subsemimodule Ç of ℬ, such that ℬ= Ų ⨁ Ç and ℬ is called a direct sum of Ų and Ç.

Definition 10. (5).
A subsemimodule Ų of an ℛsemimodule ℬ is called maximal subsemimodule of ℬ if it is not contained properly in any other proper subsemimodule of ℬ.

Definition 12. (10).
A semimodule ℬ is called uniform semimodule if the intersection of any nonzero two subtractive subsemimodules of ℬ is nonzero subsemimodule of ℬ. Definition 13. (5). Let ℬ be a semimodule and b∈ ℬ . The left annihilator of b is defined by monomorphism if is one-one and it is isomorphism if is one-one and onto. Also we denoted ℛ (ℬ) is the set Ș of endomorphisms of ℬ.
In (6) a generalization for injective modules were given, in (3) the following concept was given analogous to that concept for semimodules.

The Jacobson radical of endomorphism semiring and some related concepts
In this part we study the Jacobson radical of endomorphism semiring, in particular for P.Q.injective semimodule and some related concepts. We add some remarks that help us to avoid some problems which we encountered. For Ș = ℛ (ℬ), we discuss (Ș), Soc(ℬ), W(Ș). We introduce the concept of kasch semimodule, and study their relationship with Jacobson radical, also, C 2condition, where semimodule is said to satisfy the C 2 -condition if every subsemimodule of ℬ which is isomorphic to a direct summand of ℬ is itself a direct summand of ℬ, and principally selfgenerator, an ℛ-semimodule ℬ is said to be principally self-generator if for every element b∈ ℬ, there exists an epimorphism : ℬℛ . These concepts are mentioned for modules in (6) and (12).
Since J + ℛ =ℛ and b∈ Rad(ℛ) ⊂ J, a contradiction hence Į = ℛ, consequently ℛ is small. //// In (13) the notion ‫״‬P-injective ring‫״‬ was introduced, where the ring ℛ is said to be Pinjective if R ℛ is a P.Q.-injective module. Analogous, we introduce this notion for semiring as follows, a semiring ℛ is called P-injective if R ℛ is a P.Q.-injective semimodule.
Lemma 16. If ℬ is subtractive, semisubtractive and cancellative semimodule, then a subsemimodule Ų of ℬ is a direct summand of ℬ if and only if the inclusion map i:Ų → ℬ has left inverse.