# Ring Theory

Ring Theory is the study of associative rings. An associative ring is an Abelian group with a second multiplication * which is distributive over the addition, and so that (*a***b*)**c*=*a**(*b***c*) for all *a*, *b*, *c* in the Abelian group. Various associative rings arise across Mathematics, for instance, group rings in Algebra, rings of smooth functions in Geometry, rings of differential operators in PDEs, Hopf algebras in Mathematical Physics.

Many properties of associative rings can be studied in general. Homological Algebra and Deformation Theory are subjects where one narrows down properties of rings while looking at rather general algebras. In Commutative Rings and Hopf Algebras one restricts a class of algebras considered while remaining inquisitive about general properties. In Group Rings, Hecke Algebras, Quantum Groups and Temperley-Lieb Algebras one narrows down a class of rings to only those obtained by a very specific construction. In Grobner-Shirshov Bases one employs a powerful algorithmic method to study different classes of rings.

Fields of interest include:

- Commutative Rings
- Deformation Theory
- Grobner-Shirshov Bases
- Group Rings
- Hecke Algebras
- Hopf Algebras
- Quantum Groups
- Temperley-Lieb Algebras