Commutative Bass rings, which form a special class of Gorenstein rings, have
been thoroughly investigated by Bass [1]. The definitions do
not carry over to non-commutative rings. However, in case one deals with
orders in separable algebras over fields, Bass orders can be defined. Drozd,
Kiricenko, and Roïter [3] and Roïter [6] have
clarified the structure of Bass orders, and they have classified them. These
Bass orders play a key role in the question of the finiteness of the
non-isomorphic indecomposable lattices over orders (cf. [2;
8]). We shall use the results of Drozd, Kiricenko, and Roïter
[3] to compute the Grothendieck groups of Bass orders
locally. Locally, the Grothendieck group of a Bass order (with the exception
of one class of Bass orders) is the epimorphic image of the direct sum of
the Grothendieck groups of the maximal orders containing it.
The Grothendieck Group of an n-exangulated Category
