Free solvable P-algebras

The construction of a free solvable P-algebra of finite degree k in the var iety of all solvable P-algebras of degree at most k (k greater than or equa l to 1) has been given. Some properties of the same have been studied. The structure of the free solvable P-algebra has been viewed as a module over a ring with several objects. The Magnus embedding theorem associated with th e Fox-derivative in a free group ring has been considered to prove properti es associated with the partial (Fox) derivative in a free associative ring. Residual nilpotency and triviality of the center of a free metabelian P-al gebra has been proved. Various properties of a homomorphism associated with a free metabelian P-algebra of finite rank have been studied. The non-embe dding property of a free solvable P-algebra of degree k of higher rank in a lower rank has also been presented here.