## Basic Exercise Problems in Module Theory

## Problem 408

Let $R$ be a ring with $1$ and $M$ be a left $R$-module.

**(a)** Prove that $0_Rm=0_M$ for all $m \in M$.

Here $0_R$ is the zero element in the ring $R$ and $0_M$ is the zero element in the module $M$, that is, the identity element of the additive group $M$.

To simplify the notations, we ignore the subscripts and simply write

\[0m=0.\]
You must be able to and must judge which zero elements are used from the context.

**(b) **Prove that $r0=0$ for all $s\in R$. Here both zeros are $0_M$.

**(c)** Prove that $(-1)m=-m$ for all $m \in M$.

**(d)** Assume that $rm=0$ for some $r\in R$ and some nonzero element $m\in M$. Prove that $r$ does not have a left inverse.