On Various 2-absorbing prime ideals in non commutative rings

In this paper we analyze strongly 2-absorbing prime ideals (shortly strongly 2-API), strongly 2-absorbing weak prime ideals (shortly strongly 2-AWPI) and 2-absorbing weak prime ideals (shortly 2-AWPI) in a non-commutative ring, which represent generalization of prime ideals (shortly PI) in a non-commutative ring. The relationship between the strongly 2-API and the 2-absorbing prime ideal (shortly 2-API) is examined. We provide examples to illustrate the new concept of strongly m_a1-system and strongly m_a2-system as well as the relationships beween them. Let I be an ideal of 𝒭 and 𝒨 be a strongly m_a1-system such that I ∩ 𝒨 = ϕ. Then there exists a strongly 2-API 𝒫 of 𝒭 containing I such that 𝒫 ∩ 𝒨 = ϕ. We prove that 𝒫 is a strongly 2-API if and only if 𝒜₁𝒜₂𝒜₃ ⊆ 𝒫 implies that 𝒜₁ ⊆ 𝒫 or 𝒜₂ ⊆ 𝒫 or 𝒜₃ ⊆ 𝒫 for all ideals 𝒜₁, 𝒜₂ and A₃ of 𝒭.