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The concept of derivation in ring theory originates from mathematicians’ desire to abstract and generalize the idea of differentiation in calculus into the realm of algebra. Gottfried Wilhelm Leibniz introduced the concept of derivation and the symbol d to denote the process of differentiation, known as ”Leibniz notation”, along with a rule of differentiation in calculus referred to as the ”Leibniz rule”. In calculus, derivation is defined on the field of real numbers, which is a particular kind of ring known as a field. Every field is a prime ring, but the converse does not necessarily hold. The abstraction and generalization of derivation into algebraic structures are defined on rings in general, more specifically on prime rings. A ring derivation is a mapping from the ring to itself that satisfies the Leibniz rule. The primeness and characteristic of a ring are key prerequisites in this study. This research focuses on the First and Second Posner’s Theorems. The First Posner’s Theorem states that in any prime ring of characteristic not equal 2, equipped with two derivations such that the composition of the two derivations is also a derivation, it follows that one of the derivations must be zero. The Second Posner’s Theorem asserts that in any prime ring of characteristic not equal 2, equipped with a nonzero derivation, if for every element a in the ring, the expression ad(a)−(a)a lies in the center of the ring, then the ring is commutative. Several counterexamples are provided to reinforce the necessity of the prime ring condition and the requirement that the characteristic not be 2, as stipulated in both Posner’s Theorems. To support the First Posner’s Theorem, examples are given involving a prime ring and a non-prime ring of characteristic 2. To support the Second Posner’s Theorem, an example is provided of a non-prime, noncommutative ring with characteristic not equal to 2
FAKULTAS SAINS DAN TEKNOLOGI, UIN SUNAN KALIJAGA YOGYAKARTA
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