@phdthesis{digilib71763,
           month = {June},
           title = {BEBERAPA SIFAT DERIVASI RING PRIMA},
          school = {UIN SUNAN KALIJAGA YOGYAKARTA},
          author = {NIM.: 21106010030 Diah Ajeng Nova Ananda},
            year = {2025},
            note = {Prof. Dr. Dra. Hj. Khurul Wardati, M.Si.},
        keywords = {Derivasi Ring, Derivasi Ring Prima, Teorema Posner Pertama, Teorema Posner Kedua},
             url = {https://digilib.uin-suka.ac.id/id/eprint/71763/},
        abstract = {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}
}