Category:Definitions/Quotient Rings

This category contains definitions related to Quotient Rings.
Related results can be found in Category: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 $J$.