Definition:Quotient Ring
Definition
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 $J$.
Also denoted as
While the inline form of the fraction notation $R / J$ is usually used for a quotient ring, some presentations use the full $\dfrac R J$ form.
It is usual for the latter form to be used only when either of both of the expressions top and bottom are more complex than single symbols.
Also known as
This is also known as a factor ring.
Some sources refer to this as a residue class ring.
Also see
- Quotient Ring Addition is Well-Defined
- Quotient Ring Product is Well-Defined
- Quotient Ring is Ring
- Congruence Relation and Ideal are Equivalent
- Results about Quotient Rings can be found here.
Linguistic Note
The word quotient derives from the Latin word meaning how often.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $22$. New Rings from Old
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Rings: $\S 22$. Quotient Rings: Theorem $42$
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 2.2$: Homomorphisms
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 7$: Homomorphisms and quotient algebras
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 60$. Factor rings
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): quotient ring: 1.
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): quotient ring
- 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.8$ Algebraic properties of $p$-adic integers
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): quotient ring