Integral domain
From Maths
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[hide]Definition
Given a ring (D,+,×), it is called an integral domain[1] if it is:
- A commutative ring, that is: ∀x,y∈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:
- ∃c∈R[c≠e+∧ac=e+]or if (by writing e+ as 0 we can say: ∃c∈R[c≠0∧ac=0])
- ∃d∈R[d≠e+∧da=e+](by writing e+ as 0 we can say: ∃d∈R[d≠0∧da=0])
- We can write this as: ∃c∈R[c≠0∧(ac=0∨ca=0)]
- ∃c∈R[c≠e+∧ac=e+]
- An element a of a ring R is said to be a divisor of zero in R if:
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,y∈D[xy=yx]
- ∀a,b∈D[(a≠0,b≠0)⟹(ab≠0)]
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
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See also