All modules, by definition, contain the zero element, which is the additive identity of the abelian group that constitutes the module. A left R-module M over the ring R is structured as an abelian group (M, +) with an operation R × M → M, ensuring that the zero element exists within the module. However, it is crucial to distinguish that the zero element of the module M may not be the same as the zero element of the ring R.
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EV33 said:Homework Statement
I am curious if all modules contain 0.
Homework Equations
A left R-module M over the ring R consists of an abelian group (M, +) and an operation R × M → M such that certain properties hold...
The Attempt at a Solution
The definition of a module says that it is an additive group, and additive groups have the zero element. Thus, all modules contain the zero element right?
Thank you.