Let Φ be a commutative rinp.R a Φ-algebra and Φ ( X )-Φ ( X1 ,X2 ,...) the polynomial ring over Φ in non- commuting in determinates X ,X ..... A polynomial identity of R is a polynomial f(x ,...,x)€ Φ(X) such that f (a ..... a )=0 for any a , . . . ,a €R; a central polynomial of R is a polynomial f (x ,...,x) in Φ(X) 1 n such that f (a , . . . ,a )€Z(R) for any a ..... a €R , and 1 n 1 n f£(x ,...,x ) is not a polynomial identity of R. The thesis gives an account of the theory of polynomial identity algebras A central polynomial of a matrix algebra ever a commutative algebra is given. The precise relationship between the ideals of a polynomial identity algebra R and the ideals of Z(R) is stated. Central simple algebras, are also considered along with the question "under what conditions, a polynomial identity algebra becomes a Goldie ring?". For a semi simple polynomial identity algebra R, EndR is shown to be a polynomial identity algebra. If M is a finite dimensional R-module, then EndM is a polynomial identity algebra. We gave a new Proof to Kaplanskys theorem (theorem 1-3 6 ) In Chapter 2 we placed and proved two new theorems : Theorem (2-8) and theorem (2-9) •