Kekomutatifan Ring Prima terhadap Pengenol dan Multiplicative (Generalized)-Reverse Derivation
OKTAVIANI WINARSO, Iwan Ernanto, S.Si., M.Sc.
2024 | Skripsi | MATEMATIKA
Diberikan ring R dan pemetaan alfa dari R ke R. Pemetaan F dari R ke R disebut multiplicative (generalized)-reverse derivation yang berasosiasi dengan alfa jika memenuhi F(xy) = F(y)x + y(alfa)(x) untuk setiap x, y elemen di R. Selanjutnya, ring R disebut ring prima jika untuk setiap a, b elemen di R dengan aRb himpunan kosong berakibat a = 0 atau b = 0. Pada skripsi ini akan dibahas kekomutatifan ring prima R terhadap suatu kondisi yang melibatkan pengenol dan multiplicative (generalized)-reverse derivation. Secara khusus, akan ditinjau kondisi-kondisi a(F(xy) +- xy) = 0, a(F(xy) +- yx) = 0, a(F(x)F(y) +- xy) = 0, serta a(F(x)F(y) +- yx) = 0 pada ring prima R. Lebih lanjut, kondisi-kondisi ini digunakan sebagai syarat cukup ring prima R komutatif.
Let R be a ring and a mapping alpha from R to R. A mapping F from R to R associated with a mapping alpha called a multiplicative (generalized)-reverse derivation if F(xy) = F(y)x + y(alpha)(x) for all x, y elements R. Furthermore, the ring R is called a prime ring if aRb is empty set implies either a=0 or b=0 for all a, b elements R. In this undergraduate thesis, it will be discussed about the commutativity of the prime ring R for a condition involving annihilator and multiplicative (generalized)-reverse derivation. Particularly, it will investigate the conditions a(F(xy) +- xy) = 0, a(F(xy) +- yx) = 0, a(F(x)F(y) +- xy) = 0, and a(F(x)F(y) +- yx) = 0 on the prime ring R. Furthermore, these conditions are used as sufficient conditions for a commutativity of the prime ring R.
Kata Kunci : Ring Prima, Pengenol, Annihilator, Derivation, Multiplicative (Generalized)-Reverse Derivation, Komutatif