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Trivial Module


A module having only one element: the singleton set {*}. It is a module over any ring R with respect to the multiplication defined by

 a*=*
(1)

for every a in R, and the addition

 *+*=*,
(2)

which makes it a trivial additive group. The only element * is, in particular, its zero element. Therefore, a trivial module is often called the zero module, and written as {0}.

The notion of trivial module is a special case of the more general notion of trivial module structure, which can be defined on every additive Abelian group G with respect to every ring R by setting

 ag=g,
(3)

for all a in R and all g in G.


See also

Trivial, Trivial Ring, Zero Module

This entry contributed by Margherita Barile

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Cite this as:

Barile, Margherita. "Trivial Module." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/TrivialModule.html

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