A degeneration-like order for modules
Let \(A\) be a finite dimensional associative \(K\)-algebra with an identity over an algebraically closed field \(K\). It is known [see \textit{Ch. Riedtmann}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 19, 275-301 (1986; Zbl 0603.16025)] that the existence of the exact sequence \(0\to N\to M\oplus Z\to Z\to 0\) of finite dimensional \(A\)-modules implies that the module \(M\) degenerates to \(N\). The author proves that the corresponding relation \(M\leq_R N\) is a partial order on the set of isomorphism classes of finite dimensional \(A\)-modules, and this relation is in fact equivalent to the existence of a short exact sequence \(0\to Z\to Z\oplus M\to N\to 0\).
scientific article; zbMATH DE number 1292434