: This thesis is an extension (and a generalization) of the work done by V. V. Bavula., Primary Decompositions for Left Noetherian Rings, Algebr Represent Theor, 13 (2010) 103–118. Let R be a commutative Noetherian ring, M an R -module, and N a submodule of M . A primary decomposition of N is a representation of the form N = ? · · · ? where each is a primary submodule of M . If R is a commutative Noetherian ring, and M is a finitely generated R -module. Then any proper submodule N of M has a primary decomposition. In this paper, primary decomposition of a submodule of a finitely generated module over a commutative Noetherian ring was generalized for modules over a (not necessarily commutative) left Noetherian ring. First we introduce the left prime spectrum of a ring that is a natural generalization of the spectrum in the commutative situation and an analogue of associated primes, so-called associated left primes (for modules over noncommutative rings . Next we mention one other way of generalizing primary decomposition to left Noetherian rings. we say that the intersection N = a primary decomposition of N if each is a primary submodule of M . Next the notions of primary submodule and primary decomposition of a Noetherian module will be generalized to a larger family of modules. We call this family the family of uniformly finite modules. Also we introduce the shortest primary decomposition and the maximal shortest primary decomposition of a submodule of a uniformly finite module and we show that for a submodule of a uniformly finite module a shortest primary decomposition always exists, and each shortest primary decomposition is contained in a maximal shortest primary decomposition. Moreover, we introduce uniform decompositions and we show that each shortest uniform decomposition is an irredundant primary decomposition and each primary decomposition can be refined to a uniform decomposition. Then two constructions are given that describe respectively all shortest primary decompositions and all shortest uniform decompositions for left Noetherian rings. It follows that these decompositions are, in general, highly non-unique. Finally, we give a generalization of primary decomposition for left Noetherian rings for ?[M], where M is a fixed left R -module and ?[M] is the full subcategory of R -Mod. the notion of primary decomposition was generalized for submodules of ?[M]. In particular, we show that if M , and R is a commutative Noetherian ring we get the