Difference between revisions of "Integral domain"

Revision as of 05:50, 9 June 2015

Definition

Given a ring $(D,+,\times)$, it is called an integral domain^[1] if it is:

• A commutative ring, that is: $$\forall x,y\in D[xy=yx]$$
• Contains no non-zero divisors of zero
  • An element $a$ of a ring $R$ is said to be a divisor of zero in $R$ if:
    • $$\exists c\in R[c\ne e_+\wedge ac=e_+]$$ or if (by writing $e_+$ as $0$ we can say: $$\exists c\in R[c\ne 0\wedge ac=0]$$ )
    • $$\exists d\in R[d\ne e_+\wedge da=e_+]$$ (by writing $e_+$ as $0$ we can say: $$\exists d\in R[d\ne 0\wedge da=0]$$ )
    • We can write this as: $$\exists c\in R[c\ne 0\wedge(ac=0\vee ca=0)]$$

As the integral domain is commutative we don't need both $ac$ and $ca$.

Shorter definition

We can restate this as^[2] a ring $D$ is an integral domain if:

• $$\forall x,y\in D[xy=yx]$$
• $$\forall a,b\in D[(a\ne 0,b\ne 0)\implies(ab\ne 0)]$$

Example of a ring that isn't an integral domain

Consider the ring $\mathbb{Z}/6\mathbb{Z}$, the ring of integers modulo 6, notice that $[2][3]=[6]=[0]=e_+$.

This means both $[2]$ and $[3]$ are non-zero divisors of zero.

Examples of rings that are integral domains

• The integers
• $\mathbb{Z}/p\mathbb{Z}$ where $p$ is prime

See next

• Ordered integral domain

See also

• Ring
• Field

References

1. Jump up ↑ Fundamentals of Abstract Algebra - An Expanded Version - Neal H. McCoy
2. Jump up ↑ My (Alec's) own work

TODO: Cancellation laws of multiplication in a ring, page 54 Neal H McCoy - Fundamentals of Abstract Algebra is a good place to start