Category:Quotient Rings

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This category contains results about Quotient Rings.
Definitions specific to this category can be found in Definitions/Quotient Rings.


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

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


Let $R / J$ be the coset space of $R$ with respect to $J$.


Let the operation $+$ be defined on $R / J$ by addition of cosets of $J$:

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

Let us also define the operation $\circ$ on $R / J$ by by product of cosets of $J$:

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


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

Subcategories

This category has the following 4 subcategories, out of 4 total.