  We relax the definition of a string algebra to also include
infinite-dimensional algebras such as k[x,y]/(xy). Using the functorial
filtration method, which goes back to Gelfand and Ponomarev, we show that
finitely generated and artinian modules (and more generally finitely controlled
and pointwise artinian modules) are classified in terms of string and band
modules. This subsumes the known classifications of finite-dimensional modules
for string algebras and of finitely generated modules for k[x,y]/(xy). Unlike
in the finite-dimensional case, the words parameterizing string modules may be
infinite.

Rings and Algebras;Commutative Algebra