Nur Aini Hidayati, Mohammad Agung, Indriati Nurul Hidayah
A ring R is called a prime ring if aRb = (0), then a = 0 or b = 0. A prime ring is not necessarily commutative, but a prime ring R is a commutative ring if it satisfies certain properties related to symmetric biderivation. Some of them are B(x, y)oB(y, z) = 0, B(x, y)oB(y, z) = xoz, or B(x, y), oB(y, z) + xoz = 0, Vx, y, z 6/, with B is a symmetric biderivation and xoy represents the anti-commutator of x and y. In this article, we prove that If the anti-commutator is replaced with, [x, y] the commutator of x and y, the above properties is still satisfied. Therefore, in this study it will be discussed about; if/? contains a symmetric biderivation of B and/is a nonzero ideal of/?, such that it satisfies one of these forms (i) [B(x, y), B(y, z)] = 0, (ii) [B(x, y), B(y, z)] = [x, z], (iii) [B(x, y), B(y, z)] + [x, z] = 0, for all x.y.z 6/, then/? is a commutative ring. © 2022 American Institute of Physics Inc.. All rights reserved.
Mathematics Department, Universitas Negeri Malang, Malang, Indonesia