description/proof of that for ring, additive inverse of nonzero element is nonzero element
Topics
About: ring
The table of contents of this article
Starting Context
- The reader knows a definition of ring.
- The reader admits the proposition that for any group, the inverse of any non-identity element is a non-identity element.
Target Context
- The reader will have a description and a proof of the proposition that for any ring, the additive inverse of any nonzero element is a nonzero element.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(R\): \(\in \{\text{ the rings }\}\)
\(r\): \(\in R\)
//
Statements:
\(r \neq 0\)
\(\implies\)
\(- r \neq 0\)
//
2: Proof
Whole Strategy: Step 1: see that \(0\) is the identity of the group under addition and \(- r\) is the inverse of \(r\) in the group, and apply the proposition that for any group, the inverse of any non-identity element is a non-identity element.
Step 1:
Any ring is a group under addition, and \(0\) is nothing but the identity of the group.
\(- r\) is nothing but the inverse of \(r\) in the group.
By the proposition that for any group, the inverse of any non-identity element is a non-identity element, \(- r \neq 0\).
