# MA4H8 Ring Theory

**Lecturer:** Charudatta Hajarnavis

**Term(s):** Term 1

**Status for Mathematics students:** List C

**Commitment:** 30 Lectures

**Assessment:** 3 hour exam (100%). The examination paper will contain five questions of equal credit. Four questions are to be answered.

**Prerequisites: **Familiarity with basic concepts including chain conditions in rings and modules and the Artin-Wedderburn theorem, e.g. from the MA377 Rings and Modules module.

**Content**: We aim to study noncommutative rings with chain conditions. A commutative integral domain has a (unique) field of fractions. What happens if we drop the commutativity axiom? Do we now obtain a division ring of fractions? If not always then when exactly? Do we need to differentiate between the left hand side and the right hand side of the ring? Also, does the theory extend meaningfully to rings such as rings of matrices which contain zero divisors? We shall give precise answers to all these questions.

Topics covered in pursuit of the above will include prime and semiprime rings, Artinian rings, composition series, the singular submodule, Ore’s theorem leading up to Goldie’s theorems and their applications.

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

A.W.Chatters and C.R.Hajarnavis, Rings with chain conditions (QA251.5.C4)

K.R.Goodearl and R.B.Warfield,Jr., An introduction to noncommutative Noetherian Rings (QA251.5.G6)

J.C.McConnell and J.C.Robson, Noncommutative Noetherian rings (QA251.5M2)

N.H.McCoy, Rings and ideals (QA247.M33)

L.H.Rowen, Ring Theory (QA247.R68)