Difference between revisions of "Integral domain"
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Given a [[Ring|ring]] {{M|(D,+,\times)}}, it is called an ''integral domain''<ref name="FOAA">Fundamentals of Abstract Algebra - An Expanded Version - Neal H. McCoy</ref> if it is: | Given a [[Ring|ring]] {{M|(D,+,\times)}}, it is called an ''integral domain''<ref name="FOAA">Fundamentals of Abstract Algebra - An Expanded Version - Neal H. McCoy</ref> if it is: | ||
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Latest revision as of 11:09, 20 February 2016
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Contents
Definition
Given a ring [ilmath](D,+,\times)[/ilmath], it is called an integral domain[1] if it is:
- A commutative ring, that is: [math]\forall x,y\in D[xy=yx][/math]
- Contains no non-zero divisors of zero
- An element [ilmath]a[/ilmath] of a ring [ilmath]R[/ilmath] is said to be a divisor of zero in [ilmath]R[/ilmath] if:
- [math]\exists c\in R[c\ne e_+\wedge ac=e_+][/math] or if (by writing [ilmath]e_+[/ilmath] as [ilmath]0[/ilmath] we can say: [math]\exists c\in R[c\ne 0\wedge ac=0][/math])
- [math]\exists d\in R[d\ne e_+\wedge da=e_+][/math] (by writing [ilmath]e_+[/ilmath] as [ilmath]0[/ilmath] we can say: [math]\exists d\in R[d\ne 0\wedge da=0][/math])
- We can write this as: [math]\exists c\in R[c\ne 0\wedge(ac=0\vee ca=0)][/math]
- An element [ilmath]a[/ilmath] of a ring [ilmath]R[/ilmath] is said to be a divisor of zero in [ilmath]R[/ilmath] if:
As the integral domain is commutative we don't need both [ilmath]ac[/ilmath] and [ilmath]ca[/ilmath].
Shorter definition
We can restate this as[2] a ring [ilmath]D[/ilmath] is an integral domain if:
- [math]\forall x,y\in D[xy=yx][/math]
- [math]\forall a,b\in D[(a\ne 0,b\ne 0)\implies(ab\ne 0)][/math]
Example of a ring that isn't an integral domain
Consider the ring [ilmath]\mathbb{Z}/6\mathbb{Z} [/ilmath], the ring of integers modulo 6, notice that [ilmath][2][3]=[6]=[0]=e_+[/ilmath].
This means both [ilmath][2][/ilmath] and [ilmath][3][/ilmath] are non-zero divisors of zero.
Examples of rings that are integral domains
- The integers
- [ilmath]\mathbb{Z}/p\mathbb{Z} [/ilmath] where [ilmath]p[/ilmath] is prime
See next
See also
