- Thread starter
- #1

#### Bruce Wayne

##### Member

- Apr 6, 2013

- 16

**Z**

_{3}is a ring. The one portion I'd like to confirm is the additive inverse part. So here's what I'm thinking as my proof:

Given [x]

_{3}, suppose [3-x]

_{3}is the additive inverse in the set

**Z**

_{3}. Thus:

[3-x]

[3]

[0]

[0-x]

[-x]

_{3}=[3]

_{3}+ [-x]_{3}=[0]

_{3}+ [-x]_{3}=[0-x]

_{3}=[-x]

_{3}

Then, it can be shown that

[x]

[x+ -x]

_{3}+ [-x]_{3}= 0[x+ -x]

_{3}= 0

Therefore, the additive inverse condition of a ring is met for

**Z**

_{3}.