The subject material in the book is presented in Five Chapters. In the First Chapter, we study the relation between Smarandache semirings and many-valued logics, in particular Basic logic and Lukasiewicz logic. We will also definitions of Smarandache semirings, automata on BL - algebras and MV- algebra. In the Second Chapter, we study the action of Legendre's polynomials on commutative finite rings R, semirings, positively ordered commutative semirings and some other semirings. The rings Fq, Z/nZ where n is an RSA integer and commutative sub rings o f Matn (Fq) are the main examples. We study...