id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
iajs-3482	JA. Abass , Ali ; Yasin, Mohammed ; Bahjat Smeein , Shrooq ; Majeed , Abdulahman H. 	Centralizer on Lie-ideal of Semi-prime Inverse Semi-ring	2025	10	.pdf	application/pdf	4140	284	87	Proof: By hypothesis “ 𝑅(𝑘𝑞 + 𝑞𝑘) = 𝑅(𝑘)𝛼(𝑞) + 𝛼(𝑞)𝑅(𝑘) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑘, 𝑞 ∈ 𝑉” 𝑅(𝑘𝑞) + 𝑅(𝑞𝑘) = 𝑅(𝑘 + 𝑞 )𝛼(𝑘 + 𝑞) 𝑅(𝑘2 + 𝑘𝑞 + 𝑞𝑘 + 𝑞2) = 𝑅(𝑘2) + 𝑅(𝑘𝑞 + 𝑞𝑘) + 𝑅(𝑞2) = 𝑅(𝑘)𝛼(𝑘) + 𝑅(𝑘𝑞 + 𝑞𝑘) + 𝑅(𝑞)𝛼(𝑞) “𝑅(𝑘 + 𝑞)𝛼(𝑘 + 𝑞) = 𝑅(𝑘)𝛼(𝑘) + 𝑅(𝑘)𝛼(𝑞) + 𝑅(𝑞)𝛼(𝑘) + 𝑅(𝑞)𝛼(𝑞)” We get 𝑅(𝑘𝑞 + 𝑞𝑘) + 𝑅(𝑘)𝛼(𝑞)′ + 𝑅(𝑞)𝛼(𝑘 )′ = 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑘, 𝑞 ∈ 𝑉 (2)	cache/iajs-3482.pdf	txt/iajs-3482.txt