Algebraic structure is a set together with one or more binary operations that satisfy some axioms of the binary operations, for example groups, rings, and modules. Modules M is Noetherian modules if the ascending chain condition hold for every submodules of M . Let R be a ring and M mxn (R) is set of matrices which the entries are element of R , if m = n then M n (R) together with addition and multiplication matrix is a ring called matrix ring. The aim of this study is to construct the algebraic structures from the set of matrices in order to fulfill every axioms of modules. The requirements for matrix ring M n (R) and module matrix M mxn (R) that satisfy the Noetherian conditions are investigated. The Noetherian term of algebraic structures will be studied, then the algebraic structure for the entries of the matrix will be determined such that satisfy the Noetherian module. This study, it has been shown that modules M mxn (R) is satisfy Noetherian R - module if the algebraic structure (R,+) is Noetherian group and module M mxn (R) is also Artinian module. The facts regarding homomorphism modules of M mxn (R) are also obtained
