In this paper, we introduce notions of (n, m) -derivation-homomorphisms and Boolean n-derivations. Using Boolean n-derivations and m-homomorphisms, we describe structures of (n, m) -derivation-homomorphisms. | Turk J Math (2016) 40: 1374 – 1385 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Derivation-homomorphisms 1 Lingyue LI1,∗, Xiaowei XU2 Qingdao Institute of Bioenergy and Bioprocess Technology, Chinese Academy of Sciences, Qingdao, . China 2 College of Mathematics, Jilin University, Changchun, . China Received: • Accepted/Published Online: • Final Version: Abstract: In this paper, we introduce notions of (n, m) -derivation-homomorphisms and Boolean n -derivations. Using Boolean n -derivations and m -homomorphisms, we describe structures of (n, m) -derivation-homomorphisms. Key words: Derivation-homomorphism, Boolean n -derivation, (n, m) -derivation-homomorphism 1. Introduction In this paper, by a ring we shall always mean an associative ring with an identity. Homomorphisms and derivations are important in the course of researching rings. Multiderivations (., biderivation, 3-derivation, or n -derivation in general) have been explored in (semi-) rings. In 1989, Vukman [8] researched Posner’s theorems [7] for the trace map of symmetric biderivations on (semi-) prime rings. Breˇsar [1, 2] characterized biderivations on prime and semiprime rings, respectively, explaining the reason why Vukman’s results hold. In 2007, Jung and Park [3] investigated Posner’s theorems for the trace of permuting 3-derivations on prime and semiprime rings. In cases of permuting 4-derivations and symmetric n -derivations, similar results were obtained in [5] and [6]. It was proved in [10] that a skew n -derivation (n ≥ 3) on a semiprime ring R must map into the center of R . Wang et al. [9] also investigated n -derivations (n ≥ 3) on triangular algebras. In a recent paper, Li and Xu [4] described multihomomorphisms. In this paper, we consider a kind of multimapping that is either a derivation or a homomorphism for each component when the other components are .
