Acharya Nagarjuna International Journal of Mathematics & Information Technology Vol.1, No.1, pp 13-32 ANIJMIT Acharya Nagarjuna University - 2004 ON FUZZY DIMENSION OF A MODULE WITH DCC ON SUBMODULES Satyanarayana Bhavanari Department of Mathematics, Acharya Nagarjuna University, Nagarjuna Nagar – 522 510, Andhra Pradesh, INDIA. Godloza Lungisile Department of Mathematics, University of Transkei, Private Bag XI, Umtata, SOUTH AFRICA. and Mohiddin Shaw Shaik Department of Mathematics, ANU PG Centre at Ongole Ongole – 523 001, Andhra Pradesh, INDIA. (Communicated by Prof. Dr. P.V. Arunachalam) Abstract Let R be fixed (not necessarily commutative) ring with identity, and M be a unitary left R-module. Goldie [ 7 ] introduced the concept of Finite Goldie dimension (FGD, in short) in modules. Later this dimension theory was studied and developed by the authors like: Anh & Marki [ 1, 2 ]; Reddy & Satyanarayana [ 11 ]; Satyanarayana [ 12 ]; and Satyanarayana, Syam Prasad & Nagaraju [ 19 ]. After the introduction of Fuzzy set by Zadeh [ 21 ], the researchers in Mathematics were trying to introduce and study this concept of Fuzzyness in different mathematical systems under study. Fu-Zheng Pan [ 4, 5 ]; and Golan [ 6 ] were studied the concept ‘fuzzy submodule’. In the present paper, we introduced the concepts: minimal element, fuzzy linearly independent element, fuzzy basis, fuzzy dimension in modules, and proved some important theorems (Th. 3.10, Th. 4.2 and Th. 5.5) related to these concepts. Key words: Finite Goldie Dimension, uniform element, u-linearly independent element, fuzzy submodule, fuzzy dimension. AMS (2000) Subject Classification: 16D 10, 16P 60, 03E 72.