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I am studying Ring Theory from Herstein's book "Topics in algebra" and I would like to discuss one moment.

Definition 1. Let $R$ be a ring. The element $a\neq 0\in R$ is called zero-divisor if there exists $b\neq 0\in R$ such that $ab=0_R$.

Definition 2. An integral domain is the commutative ring which has not zero-divisors.

Also it is easy prove the following fact: If $D$ is an integral domain of finite characteristics then characteristics is a prime number.

Can trivial ring be an integral domain? If yes then it's characteristics is not prime which contradicts to the above fact!

But Herstein does not say nothing about triviality of integral domain.

Can anyone clarify this question, please?

Bill Dubuque
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RFZ
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    I would guess that this is an edge case that Herstein either didn't consider or didn't bother making note of (or possibly that the author doesn't consider the zero ring to be a ring - I had a professor who took that position once, although I don't think it's a great position to take). – Milo Brandt May 02 '18 at 19:58
  • Yea, generally in the definition of integral domain the ring is required to be a ring with unity $1 \neq 0$. – wgrenard May 02 '18 at 19:59
  • @wgrenard, Herstein does not require this condition. In his definition it need not to contain unity $1\neq 0$! – RFZ May 02 '18 at 20:01
  • Yea, I see what you mean. I'm just saying I agree with the comment above mine. Because every definition I've ever read requires this. I just double checked in Lang's Algebra and it specifies there that $1 \neq 0$. – wgrenard May 02 '18 at 20:04
  • You should check to see if Herstein allows the trivial ring to be included in his definition of a ring. – wgrenard May 02 '18 at 20:09
  • @wgrenard, But I am working through Herstein's book :) And i would like to know what's wrong with this case?! Unfortunately, Herstein does not say nothing about trivial ring. – RFZ May 02 '18 at 20:10
  • @RFZ It is nearly universal to say that an integral domain has nonzero identity. If one were to adopt the contrary stance, one would have to reinterpret all theorems mentioning them in other books lest they mistakenly apply a result defined the normal way. There is no obvious benefit to defining them the contrary way. – rschwieb May 02 '18 at 20:16
  • @rschwieb, you suggestion makes sense. But it's quite strange that Herstein did not care about it in his book :/ – RFZ May 02 '18 at 20:19
  • @RFZ It's hardly surprising that someone would forget to assert something they take for granted all the time. In fact, it's probably absolutely necessary for society to function that not absolutely every assumption has to be specified! I'm sure someday someone will ask you if you meant to exclude some edge case that you forgot because it seemed absurd to mention. – rschwieb May 02 '18 at 20:29

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Generally when one says characteristic is finite it means characteristic is non-zero number.

Because if you consider zero characteristic as finite case then there many non trivial rings with zero characteristic.

Eg. Ring $\mathbb Z$ has $0$ characteristic.

Mayuresh L
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  • Indeed, Herstein gives two definitions of characteristics for integral domain: the first one when it's zero and the other one when it's finite. In other words, you mean that if integral domain $D$ has finite characterics then it should not be trivial, right? – RFZ May 02 '18 at 20:13
  • Yes! That’s what above example shows. – Mayuresh L May 02 '18 at 20:17
  • I know that Ring of Integers has $0$ characteristic. But I can't get why you mention this? – RFZ May 02 '18 at 20:21
  • I mentioned this only to explain when we say finite characteristic it means its non zero finite number. – Mayuresh L May 02 '18 at 20:23
  • See also https://math.stackexchange.com/questions/98605/why-characteristic-zero-and-not-infinite-characteristic – lhf May 02 '18 at 23:46
  • Dear Mayuresh L! After some thoughts on your post i would like to ask this question: What is the characteristic of trivial ring? – RFZ May 03 '18 at 16:27
  • I am quoting this from wikipedia,“The only ring with characteristic 1 is the trivial ring, which has only a single element 0 = 1. If a nontrivial ring R does not have any nontrivial zero divisors, then its characteristic is either 0 or prime. In particular, this applies to all fields, to all integral domains, and to all division rings. Any ring of characteristic 0 is infinite”. – Mayuresh L May 03 '18 at 16:44