id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
iajs-3594	Hussein, Aya; Majeed , Abdulrahman H. ; Smeein , Shrooq Bahjat ; Abu-Shams , Azza I.M.S. 	Some Results on Double Cenralizer for Prime and Semiprime Г- rings 	2025	8	.pdf	application/pdf	3963	253	87	= 𝑇(𝑟)𝛼𝑥 = 𝑆(𝑟)𝛼𝑥 for each 𝑟, 𝑡 ∈ A , 𝑥 ∈ U and 𝛼 ∈ Г. This reduces to (𝑇(𝑟) − 𝑆(𝑟))𝛼𝑥 = 0, for each 𝑟 ∈ A , 𝑥 ∈ U and 𝛼 ∈ Г. (14) Replacing x with 𝑡𝛽𝑥 in (14), when 𝑡 ∈ A, 𝑥 ∈ U, 𝛼 ∈ Г, leads to (𝑇(𝑟) − 𝑆(𝑟))𝛼𝑡𝛽𝑥 = 0, for each 𝑟 ∈ 𝐴, 𝑥 ∈ 𝑈 𝑎𝑛𝑑 𝛼, 𝛽 ∈ Г. i.e. (𝑇(𝑟) = 𝑇(𝑥)𝛼 𝑦 , for each 𝑥, 𝑦 ∈ 𝐴 𝑎𝑛𝑑 𝛼 ∈ Г. Since T is acts homomorphism on A, thence T (x) 𝛼 T(y) = T (x) 𝛼 y, for any 𝑥, 𝑦 ∈ 𝐴 𝑎𝑛𝑑 𝛼 ∈ 𝛤. (6)	cache/iajs-3594.pdf	txt/iajs-3594.txt