description/proof of that for ring, multiple of 0 is 0
Topics
About: ring
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Proof
Starting Context
- The reader knows a definition of ring.
Target Context
- The reader will have a description and a proof of the proposition that for any ring, any multiple of 0 is 0.
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:
\(R\): \(\in \{\text{ the rings }\}\)
\(p\): \(\in R\)
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Statements:
\(0 p = p 0 = 0\).
//
2: Natural Language Description
For any ring, \(R\), and any element, \(p \in R\), \(0 p = p 0 = 0\).
3: Proof
\(0 p + p = 0 p + 1 p = (0 + 1) p = 1 p = p\), which implies that \(0 p = 0 p + p - p = p - p = 0\).
\(p 0 + p = p 0 + p 1 = p (0 + 1) = p 1 = p\), which implies that \(p 0 = p 0 + p - p = p - p = 0\).
