By an algebra Λ we mean an associative
k-algebra with identity, where k is an
algebraically closed field. All algebras are assumed to be finite dimensional
over k
(except the path algebra kQ). An algebra is said to be biserial
if every indecomposable
projective left or right Λ-module P contains
uniserial submodules U and V such that
U+V=Rad(P) and U∩V
is either zero or simple. (Recall that a module is uniserial
if it has a unique composition series, and the radical Rad(M)
of a module M is the
intersection of its maximal submodules.) Biserial algebras arose as a natural
generalization of Nakayama's generalized uniserial algebras [2].
The condition first
appeared in the work of Tachikawa [6, Proposition 2.7],
and it was formalized by
Fuller [1]. Examples include blocks of group algebras
with cyclic defect group; finite
dimensional quotients of the algebras (1)–(4) and (7)–(9)
in Ringel's list of tame local
algebras [4]; the special biserial algebras of
[5, 8] and the regularly biserial algebras
of
[3]. An algebra Λ is basic if Λ/Rad(Λ)
is
a product of copies of k. This paper contains
a natural alternative characterization of basic biserial algebras, the
concept of a
bisected presentation. Using this characterization we can prove a number
of results
about biserial algebras which were inaccessible before. In particular we
can describe
basic biserial algebras by means of quivers with relations.