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The purpose of these notes is to explain in detail some topics on the intersection of commutative algebra, representation theory and singularity theory. They are based on lectures given in Tokyo, but also contain new research. It is the first cohesive account of the area and will provide a useful synthesis of recent research for algebraists.

Cohen-Macaulay rings. --- Cohen-Macaulay modules. --- Macaulay modules, Cohen --- -Modules (Algebra) --- Macaulay local rings --- Macaulay rings, Cohen --- -Rings, Cohen-Macaulay --- Rings, Macaulay local --- Local rings --- Matsumura, Hideyuki, --- Matsumura, H. --- 松村英之, --- Ordered algebraic structures