definition of irreducible element of commutative ring
Topics
About: ring
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
Starting Context
- The reader knows a definition of ring.
- The reader knows a definition of units of ring.
Target Context
- The reader will have a definition of irreducible element of commutative ring.
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:
\( R\): \(\in \{\text{ the commutative rings }\}\)
\( U\): \(= \{\text{ the units of } R\}\)
\(*p\): \(\in R\)
//
Conditions:
\(p \neq 0\)
\(\land\)
\(p \notin U\)
\(\land\)
\(\exists p_1, p_2 \in R \text{ such that } p = p_1 p_2 \implies (p_1 \in U \lor p_2 \in U)\)
//
2: Natural Language Description
For any commutative ring, \(R\), and the set of the units of \(R\), \(U\), any element, \(p \in R\), such that if there are any \(p_1, p_2 \in R\) such that \(p = p_1 p_2\), \(p_1 \in U\) or \(p_2 \in U\)
