Sharif Digital Repository

Defining the structure of maps using local features is among the popular fields of study in mathematics. Therefore determining the structure of maps on rings which behave like derivations at idempotent-product elements has been getting attention recently. This subject is useful for examining the structure of rings and algebraic operators in both algebra and analysis as well. Suppose that R is a ring, d : R ! R is an additive map, z 2 R and d meets the condition below: 8a; b 2 R : d(ab) = ad(b) + d(a)b Therefore d is called a derivation on R. If for every a; b 2 R where ab = z, d(ab) = ad(b) + d(a)b then d behaves like a derivation at idempotent-product elements of ab = z. The main challenge... 

Our main purpose is introducing essential dimension and investigating properties of this concept and definition of it on different algebraic objects and proving some theories about it. In the beginning we define the concept of essential dimension on the extension fields which indeed it is expressing complexity of extension on the background field.Then with the meaning of noetherian extension which we will introduce it in the chapter three, we generalize the concept of essential dimension to finite groups. At last we investigate the connection between the essential dimension with generic polynomials and one of our important results is finding upper bounds for essential dimension of finite...