Noncommutative Rings

We restricted to commutative unital rings for the previous pages, now let us have a look at some noncommutative rings. In particular, the associative algebras.

Ideals in Non-Commutative Rings

Ideals in Non-Commutative Rings

When the ring is non-commutative we have three variants of an ideal: an abelian subgroup in a ring is

• left ideal if for all and
• right ideal if for all and
• a two-sided ideal if it is both left and right.
  

e.g. In the left (right) ideals are matrices having some columns (rows) consisting of zeros and the other columns (rows) arbitrary. There are no non-trivial two-sided ideals in , it is a simple ring.

Associative Algebras

Algebra

An algebra is a (possibly non-commutative) ring equipped with a vector space structure over a field such that the multiplication is compatible with the scalar multiplication. That is, the ring multiplication is bilinear: for all and we have The algebra is called non-associative if the multiplication is non-associative. It is unital if there is an element such that for all .

e.g. Matrices with entries in some field , with scalar multiplication and matrix multiplication are a prominent example of an algebra. Polynomials with scalar and polynomial multiplication are another. One example which will become an important prototype for the structures we study in this course is an algebra constructed from an arbitrary group, called the group algebra:

Group Algebra

Given a group and a field , the group algebra of over , as a vector space, is given by formal linear combinations of elements of : that is, the vector space with basis . The algebra multiplication given by the bilinear extension of multiplication in .

e.g. The quaternions is the subring (-subalgebra) of generated by the matrices:

Proposition

A group algebra is isomorphic to finitely supported ‑valued functions on with convolution product.

Proof We can identify each with a function defined by for and for all other . Under this identification, the multiplication in corresponds to the convolution product of functions:

Simple and Semisimple Algebra

An algebra is simple, if it has no nontrivial ideals;
is semisimple if has no nontrivial nilpotent ideals.