= Ring theory
Ring theory studies <rings>, their <ideals>, <modules>, and structure-preserving <ring homomorphisms>. It includes both <commutative algebra> and <noncommutative algebra>. Here rings are associative and have an identity unless another convention is specified.
The <integers> are a commutative example. Matrix rings are generally noncommutative: in $M_2(k)$, $E_{12}E_{21}=E_{11}$ while $E_{21}E_{12}=E_{22}$. Thus techniques that reorder scalar factors do not automatically work for matrices.
<Quotient rings> organize the information lost by a <ring homomorphism>. If $f:R\to S$, the <first isomorphism theorem for rings> identifies $R/\ker f$ with the image of $f$. Studying <ideals> describes such quotients, while studying <modules> describes how a <ring> acts by additive transformations. The two viewpoints connect arithmetic examples with matrix and operator examples.
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