Integral domain

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Definition

Given a ring (D,+,×), it is called an integral domain[1] if it is:

  • A commutative ring, that is: x,yD[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:
      • cR[ce+ac=e+]
        or if (by writing e+ as 0 we can say: cR[c0ac=0]
        )
      • dR[de+da=e+]
        (by writing e+ as 0 we can say: dR[d0da=0]
        )
      • We can write this as: cR[c0(ac=0ca=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:

  • x,yD[xy=yx]
  • a,bD[(a0,b0)(ab0)]

Example of a ring that isn't an integral domain

Consider the ring Z/6Z, 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
  • Z/pZ where p is prime

See next

See also


References

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