Exploring if 0 is a Prime Element in an Integral Domain

In summary, the conversation revolves around the question of whether 0 is a prime element in an integral domain. The definition of a prime element is discussed, which includes the condition of being nonzero. It is pointed out that in a ring, all elements can be divided into three categories: zero divisors, units, and other elements. Primes fall under the category of other elements, and in good cases, all irreducibles are prime. Ultimately, it is concluded that 0 is not a prime element based on the definition and characteristics of primes.
  • #1
quasar987
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[SOLVED] Is 0 a prime?

Am I missing something or is 0 a prime element in an integral domain?

In the definition of prime element p of an integral domain, we only ask that the ideal generated by p, be prime.

Well (0) is obviously prime because if ab=0 in an integral domain, then it is that a=0 or b=0.
 
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  • #2
What are the factors of zero?

It might or might not be composite, but it certainly is NOT prime.
 
  • #3
I don't understand the point you're making.
 
  • #4
I'm not sure on this but one definition I've heard for a prime is “a number that may only be evenly divided by one and itself.” Zero can be divided by one but dividing by itself would be dividing by zero and hence would be an undefined result.
 
  • #5
Hi CQ and thanks for your input.

But as far as I can see, this is a definition that makes sense only for integers (and possibly in Euclidean Domains). My point is precisely that the general definition seems to be is in contradiction with this one.

But then again, the general definition claims that -2,-3,-5,... are also primes in Z, while the classical definition considers only positive primes. So it could very well be that 0 is prime in the general setting.

I just wanted to make sure because it surprised me a little to come to this conclusion.
 
  • #6
Hmm… “a number that may only be evenly divided by one and itself and its additive inverse” then, maybe?

Or alternatively it could simply be that zero is an uninteresting prime because it's never a factor of another number.

MathWorld has http://mathworld.wolfram.com/PrimeNumber.html" on prime numbers, BTW.
 
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  • #7
quasar987 said:
Am I missing something or is 0 a prime element in an integral domain?

In the definition of prime element p of an integral domain, we only ask that the ideal generated by p, be prime.

Well (0) is obviously prime because if ab=0 in an integral domain, then it is that a=0 or b=0.

no definitely not, the definition of a prime element b in a ring R , is that b is a nonzero nonunit element s.t. b |ac => b |a or b| c
 
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  • #8
Oh yes, I missed the "nonzero" part in the definition. Funny because I re-read it just before posting too.
 
  • #9
all ring elements are divided into three classes: zero divisors (including zero), units (including 1), and the other elements.

included among these other elements are the irreducibles, among which are the primes.

in good cases, all irreducibles are prime, and all these other elements are products of primes.

so primes are never units and never zero divisors, and vice versa.
 

Related to Exploring if 0 is a Prime Element in an Integral Domain

1. Is 0 considered a prime element in an integral domain?

No, 0 is not considered a prime element in an integral domain. In fact, it is defined as a non-unit, meaning it cannot be inverted to obtain an element with a multiplicative inverse. Additionally, the fundamental theorem of arithmetic states that every non-zero integer can be uniquely expressed as a product of primes, and since 0 cannot be expressed as a product of primes, it cannot be considered a prime element.

2. Can 0 be divided by any other element in an integral domain?

No, 0 cannot be divided by any other element in an integral domain. This is because division in an integral domain is defined as the multiplication by the multiplicative inverse of the divisor. Since 0 does not have a multiplicative inverse, division by 0 is undefined.

3. How does the concept of prime elements apply to 0 in an integral domain?

The concept of prime elements does not apply to 0 in an integral domain. Prime elements are defined as non-zero elements that cannot be factored into smaller non-unit elements. Since 0 is not considered a prime element, it cannot be factored in this way.

4. Are there any exceptions to 0 not being a prime element in an integral domain?

No, there are no exceptions to 0 not being a prime element in an integral domain. This is a fundamental concept in mathematics and is accepted universally.

5. What is the significance of exploring if 0 is a prime element in an integral domain?

The significance of exploring this concept lies in understanding the fundamental properties of integral domains and prime elements. It also helps to clarify any misconceptions or confusion surrounding the definition of prime elements and their application in different mathematical contexts.

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