Sufficient condition of symmetric biderivation on prime ring to be commutative ring

Abstract

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, ∀ x, y, z ∈ I, 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 R contains a symmetric biderivation of B and I is a nonzero ideal of R, 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 ∈ I, then R is a commutative ring.