On Various 2-absorbing prime ideals in non commutative rings

Abstract

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 ma1-system and strongly ma2-system as well as the relationships beween them. Let I be an ideal of 𝒭 and 𝒨 be a strongly ma1-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 𝒭.

Keywords

• strongly 2-api
• strongly 2-awpi
• 2-awpi
• strongly ma1-system
• strongly ma2-system
• weak ma1-system
• weak ma2-system
• primary 05c25, 06b10, 16d25