Lie Ideals and Homoderivations in Semiprime Rings

Ali Yahya Hummdi, Zeliha Bedir, Emine Koç Sögütcü, Öznur Gölbaşı and Nadeem ur Rehman ()
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Ali Yahya Hummdi: Department of Mathematics, College of Science, King Khalid University, Abha 61471, Saudi Arabia
Zeliha Bedir: Department of Mathematics, Faculty of Science, Sivas Cumhuriyet University, Sivas 58140, Turkey
Emine Koç Sögütcü: Department of Mathematics, Faculty of Science, Sivas Cumhuriyet University, Sivas 58140, Turkey
Öznur Gölbaşı: Department of Mathematics, Faculty of Science, Sivas Cumhuriyet University, Sivas 58140, Turkey
Nadeem ur Rehman: Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

Mathematics, 2025, vol. 13, issue 4, 1-13

Abstract: Let S be a 2-torsion free semiprime ring and U be a noncentral square-closed Lie ideal of S . An additive mapping ℏ on S is defined as a homoderivation if ℏ ( a b ) = ℏ ( a ) ℏ ( b ) + ℏ ( a ) b + a ℏ ( a ) for all a , b ∈ S . In the present paper, we shall prove that ℏ is a commuting map on U if any one of the following holds: (i) ℏ ( a ˜ 1 a ˜ 2 ) + a ˜ 1 a ˜ 2 ∈ Z , (ii) ℏ ( a ˜ 1 a ˜ 2 ) − a ˜ 1 a ˜ 2 ∈ Z , (iii) ℏ a ˜ 1 ∘ a ˜ 2 = 0 , (iv) ℏ a ˜ 1 ∘ a ˜ 2 = a ˜ 1 , a ˜ 2 , (v) ℏ a ˜ 1 , a ˜ 2 = 0 , (vi) ℏ a ˜ 1 , a ˜ 2 = ( a ˜ 1 ∘ a ˜ 2 ) , (vii) a ˜ 1 ℏ ( a ˜ 2 ) ± a ˜ 1 a ˜ 2 ∈ Z , (viii) a ˜ 1 ℏ ( a ˜ 2 ) ± a ˜ 2 a ˜ 1 = 0 , (ix) a ˜ 1 ℏ ( a ˜ 2 ) ± a ˜ 1 ∘ a ˜ 2 = 0 , (x) [ ℏ ( a ˜ 1 ) , a ˜ 2 ] ± a ˜ 1 a ˜ 2 = 0 , (xi) [ ℏ ( a ˜ 1 ) , a ˜ 2 ] ± a ˜ 2 a ˜ 1 = 0 , for all a ˜ 1 , a ˜ 2 ∈ U , where ℏ is a homoderivation on S .

Keywords: semiprime ring; Lie ideal; derivation; homoderivation; commutativity (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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