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

<The previous article in this series | The table of contents of this series | The next article in this series>

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:
$R$: $\in \{\text{ the commutative rings }\}$
$I$: $\in \{\text{ the both-sided ideals of }R\}$
$R/I$: $=\text{ the quotient ring }$
//

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:

$[p_1][p_2]=[p_1{p}_2]=[p_2{p}_1]=[p_2][p_1]$.

---

References

<The previous article in this series | The table of contents of this series | The next article in this series>