Every ring with unity has at least two units

[Mr Davis 97]

Homework Statement

T/F: Every ring with unity has at least two units

Homework Equations

The Attempt at a Solution

I thought that the answer was true, because if a ring $R$ has unity $1$, then $1 \cdot 1 = 1$ and $(-1) \cdot (-1) = 1$. Where am I going wrong?

 

[Stephen Tashi]

I thought that the answer was true, because if a ring $R$ has unity $1$, then $1 \cdot 1 = 1$ and $(-1) \cdot (-1) = 1$. Where am I going wrong?

The elements denoted by "$a$" and "$-a$" are not necessarily distinct from each other. Can you think of a ring where 1 = -1 ?

 

[Mr Davis 97]

The elements denoted by "$a$" and "$-a$" are not necessarily distinct from each other. Can you think of a ring where 1 = -1 ?

I can only think of the zero ring as an example. Are there non-trivial examples?

 

[fresh_42]

I can only think of the zero ring as an example. Are there non-trivial examples?

Many. The zero ring hasn't any units at all. So what do you need also to speak of units and what is the smallest example? Based on this, there are plenty of examples.