Zero Module (original) (raw)
Every module over a ring 
contains a so-called "zero element" which fulfils the properties suggested by its name with respect to addition,
and with respect to multiplication by any element 
of 
,
This shows that the set 
is closed under both module operations, and, therefore, it itself is a module, called the zero module. It also deserves the name trivial module, since it is the simplest module possible.
See also
Module, Singleton Set, Trivial, Trivial Ring
This entry contributed by Margherita Barile
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Cite this as:
Barile, Margherita. "Zero Module." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ZeroModule.html