|Credits: 10||Convenor: Dr. R. J. Marsh||Semester: 2|
|Prerequisites:||essential: MC145, MC241|
|Assessment:||Regular Coursework: 20%||One and a half hour exam: 80%|
Much of the theory of commutative rings is motivated by the properties of the integers. This leads firstly to the idea of a Euclidean domain, which gives a general setting for the division algorithms already seen in MC145 for the integers and polynomials over a field, and secondly to the concept of a principal ideal domain, which gives particularly useful structure theorems for both rings and modules. Describing the structure of rings and modules is important in abstract algebra and enables us to apply this theory to many diverse areas of mathematics; from the examples listed above, it can be seen that the ideas in this course are used in number theory, linear algebra and group theory. The final part of this course considers some general theorems which allow us to describe the structure of any module.
To understand and be able to use the main results and proofs of this course.
To be able to investigate the properties of a ring or module.
To relate the concept of an ideal to homomorphisms and factor rings.
To distinguish between the concepts of primeness and irreducibility.
To know the interrelationships between Euclidean domains, principal ideal domains and unique factorisation domains.
To understand the unique factorisation properties motivated by the example of the integers.
To understand how every finitely generated module is a homomorphic image of a free module.
The ability to apply taught principles and concepts to new situations.
The ability to present written arguments and solutions in a coherent and logical form.
The ability to use the techniques taught within the course to solve problems.
Divisibility in an ID and its relation to principal ideals, associate, irreducible element, prime element, Z and the norm function, Euclidean domain (ED), principal ideal domain (PID), every ED is a PID, to know that there are PIDs which are not EDs, every prime element in an ID is irreducible and the converse holds in a PID, Z[x] is not a PID, unique factorisation domain (UFD), to know that every PID is a UFD and that if R is a UFD then so is R[x], discussion of unique factorisation in Z, Z and F[x] where F is a field.
Cyclic modules, characterisation of cyclic modules, finitely generated modules, direct sum construction for rings and modules, every finitely generated module is a homomorphic image of some Rn, free modules, every module is a homomorphic image of a free module.
R. B. J. T. Allenby, Rings, Fields and Groups, 2nd Ed., Arnold.
A. W. Chatters and C. R. Hajarnavis, An Introductory Course in Commutative Algebra, Oxford University Press.
J. B. Fraleigh, A First Course in Abstract Algebra, 5th Ed., Addison-Wesley.
B. Hartley and T. Hawkes, Rings, Modules and Linear Algebra, Chapman and Hall (out of print).
W. K. Nicholson, Introduction to Abstract Algebra,, PWS-Kent.
D. Sharpe, Rings and Factorization, Cambridge University Press (out of print).