Namespaces
Variants
Actions

Difference between revisions of "Quotient ring"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
Line 1: Line 1:
''of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076920/q0769201.png" /> by an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076920/q0769202.png" />''
+
{{TEX|done}}
 +
''of a ring $R$ by an ideal $I$''
  
The [[Quotient group|quotient group]] of the additive group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076920/q0769203.png" /> by the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076920/q0769204.png" />, with multiplication
+
The [[Quotient group|quotient group]] of the additive group of $R$ by the subgroup $I$, with multiplication
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076920/q0769205.png" /></td> </tr></table>
+
$$(a+I)(b+I)=ab+I.$$
  
The quotient turns out to be a ring and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076920/q0769206.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076920/q0769207.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076920/q0769208.png" />, is a surjective ring homomorphism, called the natural homomorphism (see [[Algebraic system|Algebraic system]]).
+
The quotient turns out to be a ring and is denoted by $R/I$. The mapping $\pi:R\to R/I$, where $\pi(x)=x+I$, is a surjective ring homomorphism, called the natural homomorphism (see [[Algebraic system|Algebraic system]]).
  
The most important example of a quotient ring is the ring of residues modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076920/q07692010.png" /> — the quotient ring of the ring of integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076920/q07692011.png" /> by the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076920/q07692012.png" />. The elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076920/q07692013.png" /> can be assumed to be the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076920/q07692014.png" />, where the sum and the product are defined as the remainders on diving the usual sum and product by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076920/q07692015.png" />. One can establish a one-to-one order-preserving correspondence between the ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076920/q07692016.png" /> and the ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076920/q07692017.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076920/q07692018.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076920/q07692019.png" /> is simple (cf. [[Simple ring|Simple ring]]) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076920/q07692020.png" /> is a [[Maximal ideal|maximal ideal]].
+
The most important example of a quotient ring is the ring of residues modulo $n$ — the quotient ring of the ring of integers $\mathbf Z$ by the ideal $\mathbf Zn$. The elements of $\mathbf Z/\mathbf Zn$ can be assumed to be the numbers $\{0,\ldots,n-1\}$, where the sum and the product are defined as the remainders on diving the usual sum and product by $n$. One can establish a one-to-one order-preserving correspondence between the ideals of $R/I$ and the ideals of $R$ containing $I$. In particular, $R/I$ is simple (cf. [[Simple ring|Simple ring]]) if and only if $I$ is a [[Maximal ideal|maximal ideal]].
  
  
  
 
====Comments====
 
====Comments====
Another most important example is the quotient ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076920/q07692021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076920/q07692022.png" /> is the ring of polynomials over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076920/q07692023.png" /> in one variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076920/q07692024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076920/q07692025.png" /> is an [[Irreducible polynomial|irreducible polynomial]]. This quotient ring describes all field extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076920/q07692026.png" /> by roots of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076920/q07692027.png" /> (cf. also [[Extension of a field|Extension of a field]]).
+
Another most important example is the quotient ring $F[x]/F[x]f(x)$, where $F[x]$ is the ring of polynomials over $F$ in one variable $x$ and $f(x)$ is an [[Irreducible polynomial|irreducible polynomial]]. This quotient ring describes all field extensions of $F$ by roots of the equation $f(x)=0$ (cf. also [[Extension of a field|Extension of a field]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,  "Algebra" , '''1''' , Wiley  (1982)  pp. Sect. 10.1</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,  "Algebra" , '''1''' , Wiley  (1982)  pp. Sect. 10.1</TD></TR></table>

Revision as of 19:51, 11 April 2014

of a ring $R$ by an ideal $I$

The quotient group of the additive group of $R$ by the subgroup $I$, with multiplication

$$(a+I)(b+I)=ab+I.$$

The quotient turns out to be a ring and is denoted by $R/I$. The mapping $\pi:R\to R/I$, where $\pi(x)=x+I$, is a surjective ring homomorphism, called the natural homomorphism (see Algebraic system).

The most important example of a quotient ring is the ring of residues modulo $n$ — the quotient ring of the ring of integers $\mathbf Z$ by the ideal $\mathbf Zn$. The elements of $\mathbf Z/\mathbf Zn$ can be assumed to be the numbers $\{0,\ldots,n-1\}$, where the sum and the product are defined as the remainders on diving the usual sum and product by $n$. One can establish a one-to-one order-preserving correspondence between the ideals of $R/I$ and the ideals of $R$ containing $I$. In particular, $R/I$ is simple (cf. Simple ring) if and only if $I$ is a maximal ideal.


Comments

Another most important example is the quotient ring $F[x]/F[x]f(x)$, where $F[x]$ is the ring of polynomials over $F$ in one variable $x$ and $f(x)$ is an irreducible polynomial. This quotient ring describes all field extensions of $F$ by roots of the equation $f(x)=0$ (cf. also Extension of a field).

References

[a1] P.M. Cohn, "Algebra" , 1 , Wiley (1982) pp. Sect. 10.1
How to Cite This Entry:
Quotient ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quotient_ring&oldid=16376
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article