Entry requirements

5 credits in mathematics. Participation in Algebra I.

Learning outcomes

On completion of the course, the student should be able to:

• give an account of important concepts and definitions in the theory of rings;
• exemplify and interpret concepts from ring theory in specific cases;
• formulate important results and theorems covered by the course;
• describe the main features of the proofs of important theorems;
• use the theory, methods and techniques of the course to solve simple problems connected to ring theory, within for example number theory ;
• present mathematical arguments to others.

Content

The concept of ring and basic properties of rings. Important examples of rings such as the integers; rational, real and complex numbers; Gaussian integers and integers modulo n. Standard constructions of rings such as polynomial rings and power series. Examples of non-commutative rings. Different types of elements: invertible elements, irreducible elements and prime elements. Different types of rings and how they relate to each other: bodies, regions of integrity, factorial rings, principal ideal rings, Euclidean rings. Unique factorization for principal ideal rings. Partial rings, ideals, quotient rings. Homomorphism and isomorphism. Noether's first isomorphism theorem. Chinese remainder theorem as ring isomorphism.

Instruction

Lectures and problem solving sessions.

Assessment

Written (4 Credit Points) and oral (1 Credit points) examination.

If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.