Study of blow-up rings

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Commutative Algebra is one of the branches of algebra, which works on the sets equipped with two binary operations, such that addition and multiplication, such as the sets of numbers and the sets of functions. The purpose is to explore the structure of these sets with view-point from two binary operations. This is a very abstract approach but it contains good applications, like cryptography, to real societies. I have posed three concrete problems and have been trying to solve them. I obtain rather excellent and big results, which I have reported in several conferences of abroad.