Direct sum of π-projective semimodules

Abstract

Let A, and N are a semiring ,and a left A- semimodule, respectively. In this work we will discuss two cases: 
 
 The direct summand of π-projective semi module is π-projective, while the direct sum of two π-projective semimodules in general is not π-projective . The details of the proof will be given.
 We will give a condition under which the direct sum of two π-projective semi modules is π-projective, as well as we also set conditions under which π-projective semi modules are projective.

On distributive semimodules

“…Proposition 3.21: [12] Let ƴ be a 𝑘 −regular homomorphism from a subtractive Ɍ-semimodule 𝒟 to Ɍsemimodule 𝒜.…”

“…i) Let Ω = {𝐵 𝑖 ∈ 𝐿(𝒟): 𝐵 𝑖 ∩ ℋ = 0} and let ℬ hence ℬ largest subsemimodule of 𝒟 which has zero intersection with ℋ and ℬ is a unique complement (see[12], Proposition 2.10). (ii) Indeed, taking 𝒜 = ∑ 𝒜 𝑖 𝑖∈𝐼 , where ℋ ≤ 𝑒 𝒜 𝑖 .…”

Distributive Semimodules: Theory, Properties, and Extensions

“…The following Proposition mentioned in [2],will be proved under different conditions. Proposition 3.5.…”

“…Let R be a commutative semiring with identity. An Rsemimodule M is said to be distributive if, for all subsemimodules 𝐴, 𝐵, and 𝐶 of 𝑀, the following equality holds: 𝐴 ∩ (𝐵 + 𝐶) = (𝐴 ∩ 𝐵) + ( 𝐴 ∩ 𝐶) [2].The notion of distributive semimodules has been studied and developed as a generalization independently in [2] and [3]. As for the module, in the last six decades much research and results on the structure of the modules with a distributive lattice of submodules (see for example [4], [5], [6], [7], and [8]).…”

“…[2] Let 𝑈 be a subtractive distributive 𝑅semimodule and 𝐵 be an 𝑅 -semimodule, and 𝜔, 𝑔 ∈ 𝐻𝑜𝑚 (𝑈, 𝐵) such that 𝜔 + 𝑔 is k-regular. Then, for any 𝐶 ∈ 𝐿(𝐵), 𝐶 = ( 𝐶 + 𝜔(𝑔 −1 (𝐶)) ∩ ( 𝐶 + 𝑔(𝜔 −1 (𝐶))).…”