Bh. Satyanarayana, D. Nagaraju, M. Babu Prasad & Sk. Mohiddin Shaw International Journal of Contemporary Advanced Mathematics, (IJCM), Volume (1): Issue (2) 16 On the Dimension of the Quotient Ring R/K Where K is a Complement Satyanarayana Bhavanari bhavanari2002@yahoo.co.in Department of Mathematics, Acharya Nagarjuna University, Nagarjuna Nagar-522 510, AP, India. Nagaraju Dasari dasari.nagaraju@gmail.com Department of Science & Humanities (Mathematics Division), HITS, Hindustan University, OMR, Padur, Chennai – 603 103, India. Babu Prasad Munagala Vijayawada, Andhra Pradesh, India. Mohiddin Shaw Shaik mohiddin_shaw26@yahoo.co.in Department of Mathematics, Acharya Nagarjuna University, Nagarjuna Nagar-522 510, AP, India. Abstract The aim of the present paper is to obtain some interesting results related to the concept “finite dimension” in the theory of associative rings R with respect to two sided ideals. It is known that if an ideal H of R has finite dimension, then there exist uniform ideals U i , 1 ≤ i ≤ n of R such that the sum U 1 ⊕ U 2 ⊕ … ⊕ U n is essential in H. This n is independent of choice of uniform ideals and we call it as dimension of H (we write dim H, in short). We obtain some important relations between the concepts complement ideals and essential ideals. Finally, we proved that dim(R/K) = dim R – dim K for a complement ideal K of R. We include some necessary examples. Keywords: Ring, Two Sided Ideal, Essential Ideal, Uniform Ideal, Finite Dimension, Complement Ideal. 1. INTRODUCTION The dimension of a vector space is defined as the number of elements in the basis. One can define a basis of a vector space as a maximal set of linearly independent vectors or a minimal set of vectors which span the space. The former when generalized to modules over rings becomes the concept of Goldie dimension. Goldie proved a structure theorem for modules which states that “a module with finite Goldie dimension (FGD, in short) contains a finite number of uniform submodules U 1 , U 2 , …, U n whose sum is direct and essential in M”. The number n obtained here is independent of the choice of U 1 , U 2 , …, U n and it is called as Goldie dimension of M. The concept Goldie dimension in Modules was studied by several authors like Satyanarayana, Mohiddin Shaw.