---- > [!definition] Definition. ([[split short exact sequence of modules]]) > > Let $R$ be a [[ring]]. A particular case of [[short exact sequence|short exact sequences]] of $R$-[[module|modules]] arises by considering the second projection from a [[direct sum of modules|direct sum]] $M_{1} \oplus M_{2} \to M_{2}$; there is then an [[exact sequence]] $0 \xrightarrow{} M_{1} \hookrightarrow M_{1} \oplus M_{2} \twoheadrightarrow M_{2} \to 0$ > obtained by identifying $M_{1}$ with the [[kernel of a module homomorphism|kernel]] of the projection. A [[short exact sequence]] $0 \to M_{1} \to N \to M_{2} \to 0$ > is said to **split** if is [[isomorphism|isomorphic]] to one of these sequences, in the sense that there is a commutative diagram (a [[chain map]]) > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRGJAF9T1Nd9CKAIzkqtRizYBZAPpCuPEBmx4CRAEyjq9Zq0QgAcgt4qBRAMxbxu6TPXGlfVYOQAWKzsn6O3E-zUoZEJinnrsDsr+LiLB2hJhskIA5BFOZiiasdZeIIlJAAQAOoUQaMxw+bLqKb6OpgHIllmhttWp9S7uzfFsPmIwUADm8ESgAGYAThAAtkhkIDgQSEK1kzPL1ItI6qtTs4iaC0uI5rvrJ5vHrmf7AKyXSABsN08PiADsm3RYDGwAFhAIABrBxrfafI5IAAc1D+MDoUDYOAA7hA4QiEC9EDDIYgAJxYkS4x5xGz6YrYWZYw5bD6knIUrBUxRgpCWXE4lrkwqUrgUThAA > \begin{tikzcd} > 0 \arrow[r] & M_1 \arrow[r] \arrow[d, "\sim"] & N \arrow[r] \arrow[d, "\sim"] & M_2 \arrow[r] \arrow[d, "\sim"] & 0 \\ > 0 \arrow[r] & M_1' \arrow[r, hook] & M_1' \oplus M_2' \arrow[r, two heads] & M_2' \arrow[r] & 0 > \end{tikzcd} > \end{document} > ``` > in which the vertical maps are all [[module isomorphism|module isomorphisms]]. [^1] > ^definition > [!equivalence] > See [[the splitting lemma]]. ^equivalence > [!basicnonexample] > ^nonexample The sequence ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```