Modules over a ring are a generalization of abelian groups (which are modules over Z). They are basic in the further study of algebra. Section 1 is mostly devoted to carrying over to modules various concepts and results of group theory. Although the classification (up to isomorphism) of modules over an arbitrary ring is quite difficult, we do have substantially complete results for free modules over a ring (Section 2) and finitely generated modules over a principal ideal domain (Section 6). Free modules, of which vector spaces over a division ring are a special case, have widespread applications and are studied thoroughly in Section 2. Projective modules (a generalization of free modules) are considered in Section 3; this material is needed only in Section VIII.6 and Chapter IX.
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