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Treballs finals del Màster en Matemàtica Avançada, Facultat de Matemàtiques, Universitat de Barcelona: Curs: 2023-2024.  Director: Joan Elías García

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The aim of this project is to study Artin rings which are fundamental structures which arise in broad areas of mathematics including algebraic geometry number theory and representation theory and therefore studying and classifying them can give new and deep perspectives for solving problems in many different areas.

In this thesis we start by reviewing the preliminaries to establish the Matlis duality which was introduced in [11] which was closely related to the work of Francis Sowerby Macaulay. Macaulay established a correspondence between Gorenstein Artin algebras $A=R / I$ and cyclic submodule $\langle F\rangle$ of the polynomial where $R$ is the power series ring in n variable and $S$ is polynomial ring with the module structure of $S$ depending on the characteristic of the given field. This correspondence can be seen as special case of the Matlis duality because the injective hull of $\mathbf{k}$ as $R$ module is isomorphic to $S$.