# Definition of an Integral Domain

In Herstein's Topics in Algebra, the definition of integral domain is stated as commutative ring without zero divisors. While i'm reading in some other books, the definition is given as commutative ring with unity without zero divisors. I little bit confused: which one should be right? I just stucked in this in example: Set of all even integers 2Z is a ring without unity having no zero divisors. If we seen by the definition of Herstein, it is integral domain. if we gone for some other text it is not

• Textbooks aren't consistent on whether rings in general have a multiplicative identity: en.wikipedia.org/wiki/… – Kaj Hansen Oct 10 '16 at 2:45
• By and large integral domains are assumed to have identity (although authors don' t have to assume it if they don't want to! Looks like this is a case of that.) There is not really a notion of a "right" definition, (but of course there are varying shades of usefulness.) Just note carefully what the assumptions are, and don't be surprised in the future if there is minor variation. On occasion you will find some true omissions, and again you shouldn't be surprised by that. – rschwieb Oct 12 '16 at 15:59

I only have the Italian translation of Herstein's book at hand, but I can confirm the definition of integral domain does not require the ring to have an identity.

For instance, the proof of Lemma 3.2.2 Every finite integral domain is a field starts off by showing that a finite integral domain happens to have an identity element.

Some textbooks don't require rings to have an identity; some, but not Herstein's, require that integral domains have it. Other textbooks require all rings to have an identity. There's no consensus about this. Jacobson's Basic Algebra I uses the name rng for “rings not necessarily with identity”, and is consistent in requiring rings to have identity.

Neither book is right or wrong: they just happen to use different definitions.

The question of if a ring should be defined as having identity is widely up for debate. I assume that both of these textbooks think that you need to have identity in an Integral Domain, but one of them thinks that that is covered by saying "it's a ring" and the other doesn't.

You can confirm this by checking how each defines a ring.

• No, Herstein's book doesn't assume a ring has unity. For instance, the proof that a finite integral domain is a field starts off by proving that a finite integral domain indeed has a unity. – egreg Oct 11 '16 at 15:12
• @egreg I haven't read Herstein's book, but I find that highly surprising. Thanks for the update – Stella Biderman Oct 11 '16 at 15:13