# Category:Quotient Rings

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This category contains results about Quotient Rings.

Let $\struct {R, +, \circ}$ be a ring.

Let $J$ be an ideal of $R$.

Let $R / J$ be the (left) coset space of $R$ modulo $J$ with respect to $+$.

Define an operation $+$ on $R / J$ by:

- $\forall x, y: \paren {x + J} + \paren {y + J} := \paren {x + y} + J$

Also, define the operation $\circ$ on $R / J$ by:

- $\forall x, y: \paren {x + J} \circ \paren {y + J} := \paren {x \circ y} + J$

The algebraic structure $\struct {R / J, +, \circ}$ is called the **quotient ring of $R$ by $J$**.

## Subcategories

This category has only the following subcategory.

### M

## Pages in category "Quotient Rings"

The following 34 pages are in this category, out of 34 total.

### C

### I

### M

- Maximal Ideal iff Quotient Ring is Field
- Maximal Left and Right Ideal iff Quotient Ring is Division Ring
- Maximal Left and Right Ideal iff Quotient Ring is Division Ring/Maximal Left Ideal implies Quotient Ring is Division Ring
- Maximal Left and Right Ideal iff Quotient Ring is Division Ring/Maximal Right Ideal implies Quotient Ring is Division Ring
- Maximal Left and Right Ideal iff Quotient Ring is Division Ring/Quotient Ring is Division Ring implies Maximal Left Ideal
- Maximal Left and Right Ideal iff Quotient Ring is Division Ring/Quotient Ring is Division Ring implies Maximal Right Ideal

### Q

- Quotient Epimorphism is Epimorphism/Ring
- Quotient Ring by Null Ideal
- Quotient Ring Defined by Ring Itself is Null Ring
- Quotient Ring Epimorphism is Epimorphism
- Quotient Ring is Ring
- Quotient Ring is Ring/Quotient Ring Addition is Well-Defined
- Quotient Ring is Ring/Quotient Ring Product is Well-Defined
- Quotient Ring of Commutative Ring is Commutative
- Quotient Ring of Integers and Principal Ideal from Unity
- Quotient Ring of Integers and Zero
- Quotient Ring of Integers with Principal Ideal
- Quotient Ring of Kernel of Ring Epimorphism
- Quotient Ring of Ring with Unity is Ring with Unity