Description: In this course we study commutative rings, the natural framework for developing tools of enormously wide application in higher mathematics, including algebraic geometry, number theory and extension theory. We elaborate on a selection of topics from the first seven chapters of the classic text by Atiyah and Macdonald (see references), providing an introduction to the subject and a platform for further study and applications. The course is divided into four parts: 1. overview and introduction to commutative ring theory; 2. introduction to theory of modules, tensor products and exactness properties; 3. study of rings and modules of fractions and properties of localisation; 4. chain conditions, study of Noetherian rings, primary decompositions and seminal theorems such as the Jordan-Holder Theorem, the Hilbert Basis Theorem and the Hilbert Nullstellensatz. Given time and interests of course participants, we may explore several applications possibly together as a class or through individual or group projects. Assessment: by assignments, project work and examination. References: 1. M.F. Atiyah and I.G. Macdonald, "Introduction to Commutative Algebra", Addison-Wesley,1969. 2. M. Reid, "Undergraduate Commutative Algebra", LMS student texts, volume 29, CUP, 1995. 3. R.Y. Sharp, "Steps in Commutative Algebra", LMS student texts, volume 51, CUP 2000. |