Abstract
The purpose of this study is to obtain the commutativity of a 3-prime near ring satisfying some differential identities on Jordan ideals involving derivations and multiplicative derivations. Further, herein we discuss some examples to show the necessity of the hypothesis to our results.
Highlights
A left near ring N is a triplet (N, +, ·), where + and · are two binary operations such that (i)(N, +) is a group, (ii) (N, ·) is a semigroup, and (iii) u · (v + w) = u · v + u · w for every u, v, w ∈ N .Analogously, if instead of (iii), N satisfies the right distributive law, N is said to be a right near ring.near rings are generalized rings, need not be commutative, and most importantly, only one distributive law is postulated (e.g., Example 1.4, Pilz [1])
We show the commutativity condition for a 3-prime near ring N if any one of the following holds: (i) [d1 (u), d2 (k)] = [u, k ], (ii) d([k, u]) = [d(k), u], (iii) [d(u), k] = [u, k], (iv) d([k, u]) = d(k) ◦ u, (v) [d(k), d(u)] = 0 for all u ∈ N and k ∈ J, a Jordan ideal of N, where d, d1, d2 are derivations on N
We consider two derivations instead of one derivation, and secondly, we prove the commutativity of a 3-prime near ring N in place of a ring R in case of a Jordan ideal of N
Summary
A left near ring N is a triplet (N , +, ·), where + and · are two binary operations such that (i). If instead of (iii), N satisfies the right distributive law, N is said to be a right near ring. Near rings are generalized rings, need not be commutative, and most importantly, only one distributive law is postulated (e.g., Example 1.4, Pilz [1]). A near ring N is known as zero-symmetric if 0u = 0 for every u ∈ N (left distributive law gives that u0 = 0). N represents a zero-symmetric left near ring with Z (N ) as its multiplicative center. A near ring N is known as 2-torsion free if 2u = 0 ⇒ u = 0 for every u ∈ N.
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