= Module theory
Module theory studies <abelian groups> with <scalar multiplication> by a <ring>. It extends <linear algebra>: a <module> over a <field> is a <vector space>, while <ring> scalars let the same language describe <integer> arithmetic and <linear operators>. The new phenomena are visible before any abstract classification: some nonzero scalars cannot be divided out, some <modules> have <torsion elements of a module>, and some have no <basis of a module>.
\Image[/module-lattice-and-torsion.png]
{title=Free integer coordinates and a cyclic module with torsion}
{description=The <free module> $\mathbb Z^2$ has two independent <integer> directions. In the <cyclic module> $\mathbb Z/6\mathbb Z$, repeatedly adding $\overline1$ returns to zero after six steps: $6\overline1=\overline0$. The circle records addition by the generator; its geometry is only a drawing convention.}
The distinction is useful because an <integer> relation such as $6m=0$ carries information which disappears if one assumes all nonzero scalars have <multiplicative inverses>. A <finitely generated module> over a <principal ideal domain> can be separated into a <finite free module> and a <torsion module>; the <structure theorem for finitely generated modules over a principal ideal domain> unifies the classification of <finitely generated abelian groups> with the study of <linear operators> through <polynomial rings>. See https://kconrad.math.uconn.edu/blurbs/linmultialg/modulesoverPID.pdf[Keith Conrad's Modules over a PID, Theorem 4.1 and the operator applications].
A historical anchor is Emmy Noether's 1921 paper https://gdz.sub.uni-goettingen.de/id/PPN235181684_0083[Idealtheorie in Ringbereichen, Mathematische Annalen 83, pages 24–66]{ref}. Its first section studies <rings> and <ideals> under the condition that every <ideal> is a <finitely generated module> over its <ring>, and connects this to the <ascending chain condition>; later sections develop <primary decomposition>. This is an early source for the structural viewpoint underlying <Noetherian rings>. https://arxiv.org/abs/1401.2577[Daniel Berlyne's translation of Noether's original paper]{ref} makes those statements accessible in English.
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