Locally Finite Central Simple Algebras

We develop a comprehensive theory of algebras over a field which are locally both finite dimensional and central simple. We generalize fundamental concepts of the theory of finite dimensional central simple algebras, and introduce supernatural matrix algebras, the supernatural degree and matrix degree, and so on. We define a Brauer monoid, whose unique maximal subgroup is the classical Brauer group, and show that once infinite dimensional division algebras exist over the field, they are abundant.