On Dense Subsemimodules and Prime Semimodules

Abstract

In this paper, we study the class of prime semimodules and the related concepts, such as the class of semimodules, the class of Dedekind semidomains, the class of prime semimodules which is invariant subsemimodules of its injective hull, and the compressible semimodules. In order to make the work as complete as possible, we stated, and sometimes proved, some known results related to the above concepts.

Almost self-injective semimodules

“…It is necessary to know the basics that have been relied upon in our work, so they were clarified in this part of the work. Definition1.1 [3]. A semiring Ȑ is said to be semi-domain if ȶ ᶉ = 0, implies either ȶ = 0 or ᶉ = 0 for ȶ, ᶉ in Ȑ. Definition1.2 [1].…”

“…(4)For every element ȶ∈ Ғ, ȶ0 = 0ȶ = 0. (2) The following implications hold for any semimodule: Injective⟶ Almost-injective⟶ Almost self-injective Alwan and Alhossaini defined semi-field of fraction of the semi-domain is in [3]. Definition 2.3.…”

“…In [3] the concept semiring of quotient is discussed and studied. From the previous proposition, the following result is obtained: Corollary 2.16.…”

Dedekind Multiplication Semimodules

“…Then put R S the set of all equivalence classes of and define addition and multiplication on R S respectively by and , where also denoted by , we mean the equivalence class of . It is, then, easy to see that R S with the mentioned operations of addition and multiplication on R S in above is a semiring [3,4]. Definition (1.2): In Remark 1.1, if is the set of all not zero-divisors of .…”

“…, and is invertible. A subsemimodule of an -semimodule is called invariant subsemimodule if , , [3,12]. Definition (3.3): A semimodule is said to be duo if each subsemimodule of is invariant, [12].…”