Understanding Modules: Definition and Properties for Homework

[EV33]

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.

 

[Dick]

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.

If 'zero' means the additive identity of the group, sure.

 

[HallsofIvy]

Note that the ring, R, also has a "0" (additive identity) which is not necessarily the additive identity of M.