A module having only one element: the singleton set . It is a module over any ring with respect to the multiplication defined by
(1)
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for every , and the addition
(2)
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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 .
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 with respect to every ring by setting
(3)
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for all and all .