1

We can define integral domains as:

  1. rings without zero divisors
  2. commutative rings without zero divisors
  3. commutative rings with identity and without zero divisors

I don't know why integral domains don't have an unified definition, maybe due to historical reasons? theoretically what's the benefit to use the definition 1 instead 3 for instance?

REMARK

I noticed that when I was studying Hungerford's book and Bhattacharya's book.

Thanks in advance

user42912
  • 23,582
  • 1
    this one might help: http://math.stackexchange.com/questions/20217/why-is-an-integral-domain-a-commutative-ring-with-unity – Jonathan May 29 '13 at 14:48
  • Another thing to point out is that there's nothing wrong with having different definitions. It's not as if one of them is the one-definition-to-rule-them-all. – rschwieb May 29 '13 at 15:02

1 Answers1

3

"Why don't we all agree on a single definition for X" has the universal answer:

Different people are interested in exploring theory under different conditions, and definitions are a matter of convenience. It's natural then that one condition (like not having nonzero zero divisors) is used, it'll propagate under the same name to different schools assuming different conditions, and wind up with those extra conditions attached. (So maybe "yes, historical reasons!" is a good short answer.)

This is especially the case for "identity" and "commutativity" in ring theory. Certain schools may adopt one or the other or both, but there isn't any reason to foist either of those conventions upon other people who want to study rings without zero divisors.

rschwieb
  • 153,510