Finitely Presented Modules over Serial Rings

Abstract

Let P = ⊕ ⁿ_k=1 e _k R, Q = ⊕ ^m_l=1 e _l R be finitely generated projective modules over a serial ring R and let f: P → Q be a homomorphism. Since every homomorphism from e _k R to e _l R is given by left multiplication by an element of R _lk , therefore f is induced by left multiplication by an m × n matrix (r _ij ), where r _ij ∈ R _ij . The following lemma shows that this matrix can be diagonalized by choosing appropriate bases for P and Q.