description/proof of that quotient ring of commutative ring by ideal is 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
- 3: Proof
Starting Context
- The reader knows a definition of quotient ring of ring by ideal.
Target Context
- The reader will have a description and a proof of the proposition that the quotient ring of any commutative ring by any ideal is a 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:
\(R\): \(\in \{\text{ the commutative rings }\}\)
\(I\): \(\in \{\text{ the both-sided ideals of } R\}\)
\(R / I\): \(= \text{ the quotient ring }\)
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Statements:
\(R / I \in \{\text{ the commutative rings }\}\)
//
2: Natural Language Description
For any commutative ring, \(R\), and any both-sided ideal of \(R\), \(I\), the quotient ring, \(R / I\), is a commutative ring.
3: Proof
Whole Strategy: Step 1: take any 2 elements of \(R / I\); Step 2: compare the both orders multiplications using the commutativity of \(R\).
Step 1:
Let us take any \([p_1], [p_2] \in R / I\).
Step 2:
\([p_1] [p_2] = [p_1 p_2] = [p_2 p_1] = [p_2] [p_1]\).
