2024-12-01

877: Quotient Ring of Commutative Ring by Ideal Is Commutative Ring

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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


  • 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: { the commutative rings }
I: { the both-sided ideals of R}
R/I: = the quotient ring 
//

Statements:
R/I{ 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 [p1],[p2]R/I.

Step 2:

[p1][p2]=[p1p2]=[p2p1]=[p2][p1].


References


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