| Center |
| Faculty of Mathematics |
| Departament |
| Algebra |
| Lecturers in charge |
| Sin datos cargados |
| Met. Docent |
| In the theoretical sessions, the active part of its development will correspond basically to the professor. The participation and work of the students in the classroom will be promoted, specially in the practical sessions. |
| Met. Avaluació |
| A theoretic and practical examination will be made. In the final qualification, may also be valued the resolution of questions, problems and exposition of subjects that colud possibly be proposed. |
| Bibliografia |
| - M.F. Atiyah y I.G. Mac Donald: Introducción al álgebra conmutativa. Reverté - T.W. Hungerford: Algebra. Spriger-Verlag - N. Jacobson: Basic Algebra II. Freeman and Company - F. Kash: Modules and rings. Academic Press - H. Matsumura: Commutative algebra. Benjamin |
| Continguts |
| We begin with a revision of the general properties of unitary commutative rings: subrings, ideals, homomorphisms and quotient rings, focusing the attention on the concepts of prime and maximal ideal and their properties and characterisations. The notions of nilradical and Jacobson radical as well as the extension and contraction of ideals are studied. The first part of the subject is completed with the study of the basic module theory, focusing the attention on free modules and tensor products. Rings and modules of fractions are analised, especially the extension and contraction of ideals in rings of fractions. The last part of the subject is devoted to study of chain conditions in rings and modules: noetherian and artinian rings and length of a module and its behaviour on exact short sequences. The noetherian character of the polinomial ring over a field is also exhibited. We develop the theory of primary decomposition especially in noetherian rings. Finally, the dimension of a ring by means of ascending chain of prime ideals is introduced and studied. |
| Objetius |
| - Studying the notions of ring and module of fractions. - Studying rings and modules with chain conditons: noetherian and artinian rings and modules. - Studying the primary decomposition. |
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