Introduction

Thus far the following classes of rings have been studied or mentioned: prime, primitive, simple, semiprimitive (right) Artin, division ring, field, and commutative principal right ideal domain (3–2.4). The last four are special instances of hereditary rings. Finitely generated modules are discussed and then applied to semihereditary rings.

Aside from their own intrinsic interest, the semiprimitive Artin rings, and the right hereditary rings are fundamental building blocks in terms of which other rings are understood. Similarly, modules with a satisfactory structure theory, or modules which appear frequently are standard concepts needed to understand more general modules. Certainly, finitely generated modules fall in this class. Just as the word ‘Artinian’ refers ambiguously to either the right or left chain condition, so also by a hereditary ring usually is meant either a left or right hereditary one, and here the latter.

Hereditary rings

Throughout this chapter R is a ring with an identity element.

Definition. A ring R is right hereditary if 1 ∈ R and provided each of its right ideals is projective. A ring is right semihereditary if each finitely generated right ideal is projective.

The next lemma holds for any ring R.

LemmaLet C < N ⊕ B be modules with (C + N)/N projective. Then there exists a submodule A ≤ N ⊕ B such that C = (N ∩ C) ⊕ A and, A ≅ (C + N)/N.

Equivalently, if π: N ⊕ B → B is the natural projection, then C = (N ∩ C) ⊕ A where π(C) ≅ A < N ⊕ B.