find more info Thus, the existence of any nontrivial modules implies the existence of nontrivial modular partitions.
A special case of a decomposition of a module is a decomposition of a ring: for example, a ring is semisimple if and only if it is a direct sum (in fact a product) of matrix rings over division rings (this observation is known as the Artin–Wedderburn theorem).
As the notion of modules has been rediscovered in many areas, modules have also been called autonomous sets, homogeneous sets, stable sets, clumps, committees, externally related sets, intervals, nonsimplifiable subnetworks, and partitive sets (Brandstädt, Le Spinrad 1999).
In the figure below, vertex 1, vertices 2 through 4, vertex 5, vertices 6 and 7, and vertices 8 through 11 are a modular partition. 3
Let R be a ring.

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If Y and Z are two members of P and

u

Y

{\displaystyle u\in Y}

and

v

Z

{\displaystyle v\in Z}

, then u and v are adjacent in G if and only if Y and Z are adjacent in this quotient.
An O(n)-space alternative that matches this performance is obtained by representing the modular decomposition tree using any standard O(n) rooted-tree data structure and labeling each leaf with the vertex of G that it represents. , by Fitting’s lemma) and thus Azumaya’s theorem applies to the setup of the Krull–Schmidt theorem. 2007, Cournier Habib 1994).
Fortunately, there exists such a recursive decomposition of a graph that implicitly represents all ways of decomposing it; this is the modular decomposition. If

Y

{\displaystyle Y}

and

Z

{\displaystyle Z}

are two members of

P

{\displaystyle P}

and

u

Y

{\displaystyle u\in Y}

and

v

Z

{\displaystyle v\in Z}

, then

u

{\displaystyle u}

and

v

{\displaystyle v}

are adjacent in

G

{\displaystyle G}
click for more if and only if

Y

{\displaystyle Y}

and

Z

{\displaystyle Z}

are adjacent in this quotient.

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