Asal halkaların özel tip dönüşümlerini ihtiva eden bazı özdeşlikler
Some identities involving special types of 
mappings of prime rings
Yazar: NİHAN BAYDAR YARBİL
Danışman: PROF. DR. NURCAN ARGAÇ
Ege Üniversitesi / Fen Bilimleri Enstitüsü / Matematik Ana Bilim Dalı
Derece: Doktora, 2016
Konu: Matematik = Mathematics
Dizin: Asal halkalar = Prime rings ; Genelleştirilmiş türevler = Generalized derivatives ; Özdeşlikler = Identities

Özet:
Bu tez esas olarak bes bolumden olusmaktadr.
Birinci bolumde tez konusu tantlms olup, konu ile ilgili yap tas
niteligindeki calsmalar ozetlenmistir.
_Ikinci bolumde tezin aksna uygun olarak okunurlugu kolaylastrmak
adna baz temel tanm ve teoremlere yer verilmistir. Ayrca bir halkann
saglayabilecegi, bilinen tum ozdeslikler tantlarak, halkann yapsal ozellikleri
ile arasndaki gecise sk tutan sonuclar sunulmustur.
Ucuncu bolumde bir asal halkann degismeli olmayan bir Lie ideali
uzerinde otomorzma ihtiva eden bir diferansiyel ozdesligin sfrlayan kosullar
incelenmis ve ozdesligi olusturan donusum ile halkann yaps hakknda
elde edilen sonuclar verilmistir. Dorduncu bolum, ucuncu bolumun devam
niteligindedir. Ucuncu bolumde ele alnan diferansiyel ozdesligin merkezi olma
durumu ele alnms ve onemli sonuclar verilmistir. Ucuncu ve dorduncu bolumlerde
polinom ozdesligi ve genellestirilmis polinom ozdesligi teorileri arac olarak
kullanlmstr.
Besinci bolumde literature daha once iki farkl sekilde sunulmus olan
genellestirilmis Lie turev kavram ele alnms ve fonksiyonel ozdeslik teorisi
arac olarak kullanlmstr. Literaturde genellestirilmis Lie turev kavramna
iliskin yer alan iki tanm kapsayan bir fonksiyonel ozdeslik tanmlanms ve halka uzerinde belirli kosullar altnda bu ozdesligi olusturan donusumlerin
karakterizasyonuna ulaslmstr.
Asal halkalarn ozel tip donusumlerini ihtiva eden ozdesliklerin calslmas
ndaki amac bu ozdesligi saglayan halkann yapsyla ilgili kir sahibi olmak
veya bunun mumkun olmadg durumlarda ozdesligi olusturan donusumlerin
bicimini belirlemektir.
Tezde yer alan ispatlar polinom ozdesligi, genellestirilmis polinom ozdesligi
ve fonksiyonel ozdeslik teorilerinin kullanm yontemlerini detayl olarak sergilemekte
ve elde edilen sonuclar literaturde yer alan bir cok sonucu genellemektedir.

Abstract:
This thesis mainly consists of ve chapters.
In the rst chapter, the subject of the thesis is introduced and the
cornerstone works related to this subject are summarized.
In the second chapter, due to the 
ow of the thesis, some basic denitions
and theorems are given for facilitating of the reading process. Moreover all
well known identities, that can be satised by the ring, are introduced and the

ashing results about the transition between these notions to the structure of
the ring are given.
In the third chapter, the annihilating condition of a dierential identity
involving an automorphism on a noncommutative Lie ideal of a prime ring is
examined and some results concerning the structure of the ring and the form
of the maps involved in the identity are given. The fourth chapter is more
of the continuation of the third chapter. The central case of the dierential
identity handled in the third chapter is examined and some important results
are stated. In the third and forth chapters, both polynomial identity theory
and generalized polynomial identity theory are used as tools.
In the fth chapter, the notion of generalized Lie derivation, which is
dened in two dierent ways before, is handled and functional identity theory
is used in the proofs. A functional identity that contains two denitions related to the generalized Lie derivations is dened and the characterization of the
maps involved in this functional identity under some certain conditions on
rings are determined.
The main goal in examining the identites involving special types of
mappings of prime rings is to give the characterization of the structure of
the ring if it is possible, otherwise to determine the form of the mappings
involved in this identity.
The proofs took place in the thesis are mostly displaying the detailed
applications of poliynomial identity theory, generalized polynomial identity
theory and functional identity theory. The conclusions of the thesis are
generalizing most of the related theorems in the literature.