15727

14048

كارشناسي ارشد

جبر

اصفهان : دانشگاه صنعتي اصفهان

1399

ر، 57ص.

1399/06/10

كتابنامه

رياضي محض

رياضي

1399/06/15

2627525

Commutatively closed sets in rings Mostafa Sohrabi Varzaneh mostafasohrabi22@gmail com July 04 2020 Master of Science Thesis in Farsi Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Mohammad Reza Vedadi mrvedadi@cc iut ac irAdvisor Mr Bijan Taeri b taeri@cc iut ac ir2000 MSC 16U70 16U80Keywords Commuatively closed set Commutative closure Reversible ring Dedekind finite ringAbstract This M Sc thesis is based on the following papers Commutatively closed sets in rings communications in algebra A D L GA 2019 Let R be ring with unitary A subset S of R is called commutatively closed set if for every a b R the con dition ab S implies that ba S The set of all nilpotent elements of R is an example of a commutativelyclosed set For any non empty subset S of R a collection of subsets Si R i 0 is defined as follows S0 S and Si ab ba Si 1 It is easy to see that Si Si 1 for all i 0 The union i 0 Si is called commutative closure of S and denoted byS It is proved that a set S is commutatively closed if and only if S S An element a R is then called commu tatively closed whenever a a The collection of all commutatively closed subsets defines a topology on thering R For this topology every open set is also a closed set The known rings such as clean dedekind finite regular reversible symmetric and 2 primal rings are studied in terms of certain commutatively closed subsets For any twoelements x y R The relation x y y x is defined and shown that this relation is an equivalence Foridempotents e f R eR f R if and only if e f The class of all elements equivalent to zero is determined in Mn K where K is a field We state below further main results of the thesis proposition Let A be a subset of a ring R 1 If A is commutatively closed then its complement R A is also commutatively closed 2 A union reps an intersection of commutatively closed sets is commutatively closed 3 The collection of commutatively closed subsets defines a topology on the ring R For this topology the opensets are also closed 4 If S R with are subsets of a ring R then S S S S proposition Let R S be a ring homomorphism then a For any X R X X b If is a ring isomorphism then for any X R X X c If T S is commutatively closed in S then 1 T is closed in R d If S is reversible ker is commutatively closed e If S is Dedekind finite then 1 1 is commutatively closed proposition Let R be a unital ring and a R a commutatively closed element then 1 The element a commutes with units 2 If 2 is not a zero divisor in R then a commutes with idempotent elements 3 The element a commutes with its factors

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