Asian Journal of Applied Sciences (ISSN: 2321 – 0893) Volume 02 – Issue 05, October 2014 Asian Online Journals (www.ajouronline.com ) 591 On the Classical Primary Radical Formula and Classical Primary Subsemimodules Pairote Yiarayong 1 and Phakakorn Panpho 2 1 Department of Mathematics, Faculty of Science and Technology, Pibulsongkram Rajabhat University,Phitsanuloke 65000, Thailand E-mail: pairote0027 {at} hotmail.com 2 Faculty of Science and Technology, Pibulsongkram Rajabhat University, Phitsanuloke 65000, Thailand _________________________________________________________________________________ ABSTRACT— In this paper, we characterize the classical primary radical of subsemimodules and classical primary subsemimodules of semimodules over a commutative semirings. Furthermore we prove that if j N is a classical primary subsemimodule of , j M then j N is to satisfy the classical primary radical formula in j M if and only if 1 2 1 1 j j j n M M M N M M          is to satisfy the classical primary radical formula in . M Keywords— classical primary subsemimodule, primary subsemimodule, classical primary radical, classical primary radical formula. _________________________________________________________________________________ 1. INTRODUCTION Throughout this paper a semiring will be defined as follows: A semiring is a set R together with two binary operations called addition "  " and multiplication "  " such that   , R  is a commutative semigroup and   , R  is semigroup; connecting the two algebraic structures are the distrubutive laws :   ab c ab ac    and   a bc ac bc    for all , , . abc R  A semimodule M over a semiring R is a commutative monoid M with additive identity 0, together with a function , R M M   defined by   , rm rm such that: 1.   rm n rm rn    2.   r sm rm sm    3.     rs m r sm  4. 0 0 0 r m   5. 1 m m  for all , mn M  and , . rs R  Clearly every ring is a semiring and hence every module over a ring R is a left semimodule over a semiring . R A nonempty subset N of a R -semimodule M is called subsemimodule of M if N is closed under addition and closed under scalar multiplication. J. Saffar Ardabili S. Motmaen and A. Yousefian Darani in (2011) defined a different class of subsemimodules and called it classical prime. A proper subsemimodule N of M is said to be classical prime when for , ab R  and , m M abm N   implies that am N  or . bm N 