WEAKLY PRIMARY SUBSEMIMODULES OF PARTIAL SEMIMODULES M. SRINIVASA REDDY 1 , V. AMARENDRA BABU 2 & P. V. SRINIVASA RAO 3 1 Assistant Professor, Department of S & H, D. V. R & Dr. H. S. MIC College of Technology, Krishna, Andhra Pradesh, India 2 Assistant Professor, Department of Mathematics, Acharya Nagarjuna University, Guntur, Andhra Pradesh, India 3 Associate Professor, Department of S & H, D. V. R & Dr. H. S. MIC College of Technology, Krishna, Andhra Pradesh, India ABSTRACT The partial functions under disjoint-domain sums and functional composition is a so-ring, an algebraic structure possessing a natural partial ordering, an infinitary partial addition and a binary multiplication, subject to a set of axioms. In this paper we introduce the notion of weakly prime, weakly primary ideals in so-rings and weakly prime, weakly primary subsemimodules in partial semimodules over partial semirings and we obtain the relation between them. KEYWORDS: Weakly Prime Ideal, Weakly Primary Ideal, Weakly Prime Subsemimodule and Weakly Primary Subsemimodule INTRODUCTION The study of ) , ( D D pfn (the set of all partial functions of a set D to itself), ) , ( D D Mfn (the set of multi functions of a set D to itself) and ) , ( D D Mset (the set of all total functions of a set D to the set of all finite multi sets of D) play an important role in the theory of computer science, and to abstract these structures Manes and Benson[1] introduced the notion of sum ordered partial semirings (so-rings). In [6], we have obtained the ideal theory of so-rings. In [7], [8] we have obtained some results on partial semimodules over partial semirings and we characterize prime and semiprime subsemimodules with prime and semiprime partial ideals respectively. In this paper we obtain some characteristics of weakly prime, weakly primary ideals of so-rings and characterize weakly prime & weakly primary subsemimodules with weakly prime & weakly primary partial ideals of a partial semiring. PRELIMINARIES In this section we collect some important definitions and results for our use in this paper. Definition: [5] A partial semiring is a quadruple ) 1 ,., , (  R , Where ) , (  R is a partial monoid with partial addition ∑, ) 1 ,., (R is a monoid with multiplicative operation  and unit 1, and the additive and multiplicative structures obey the following distributive laws. If ) : ( I i x i   is defined in R, then for all y in R, ) : ( I i x y i    and ) : ( I i y x i    are defined and           i i i i i i i i y x y x x y x y ). ( ] [ ); ( ] [ Throughout this paper we consider R as a commutative partial semiring. International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN 2249-6955 Vol. 3, Issue 2, Jun 2013, 45-56 ©TJPRC Pvt. Ltd.