# MA4H8 Ring Theory

**Lecturer:** Charudatta Hajarnavis

**Term(s):** Term 1

**Status for Mathematics students:** List C

**Commitment:** 30 Lectures

**Assessment:** 3 hour exam (100%).

**Prerequisites:** Familiarity with chain conditions for modules, properties of semi-simple modules and the Artin-Wedderburn theorem from MA377 Rings and Modules is desirable. A revision will be undertaken with some alternative proofs.

**Leads To: **

**Content**: The course will be based on the lecture notes:

Both commutative and non-commutative rings will be studied. Our main aim is to develop the theory required to prove a theorem of Auslander and Buchsbaum that a (commutative) regular local ring is a unique factorisation domain. All known proofs of this theorem require methods form homological algebra. Thus we shall study properties of Noetherian rings and modules, look at projective resolutions of a module, define the global dimension of a ring and see how it relates to its Krull dimension.

**Books:** (For background reading and further study only):

M. Atiyah and I. Macdonald, Introduction to Commutative Algebra (QA 251.3.A8)

I. Kaplansky, Commutative rings (QA 251.3.K2)

J. Rotman, An Introduction to Homological Algebra (QA169.R667)

O. Zariski and P. Samuel, Commutative Algebra, vols. I & II (QA 251.3.Z2)