Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $A$ be an integral domain. Prove that if $(x+1)^2=x^2+1$ in $A[x]$, then $A$ must have characteristic $2$.

We have $x^2+2x+1=x^2+1$, so $2=0$, and hence the characteristic must be $1$ or $2$. Now, I don't see anything wrong with the characteristic being $1$. So, is the problem correct?

share|improve this question
    
An integral domain is by definition a Ring where $0$ is a prime ideal. The $0$-ring by definition doesn't have any prime ideals. (in fact it's the only commutative ring with that property) –  Louis Jan 15 at 22:06
    
@Bill: I'm using the convention of the stacks project: stacks.math.columbia.edu/tag/00AQ In my opinion a prime ideal should be a proper ideal. –  Louis Jan 15 at 22:15
1  
@Louis Indeed but one could argue otherwise in the zero ring. –  Bill Dubuque Jan 15 at 22:21
    
@Bill Can you explain from which point of view $0$ should be a prime ideal in the $0$ ring? The only thing I can imagine is the fact that we desire every ring to consider a prime ideal, this is "proven" by first showing maximal ideals exist, then that those are prime. However, this is not an axiom, but a fact that has to be proven and the proof fails in the case $A=0$. From any other point of view, I don't see a reasoning why we should regard the $0$ ideal as prime in this case. (It would also conflict the wish that we want $R/I$ to be an integral domain iff $I \subset R$ is prime) –  Louis Jan 16 at 0:26
    
I recognise you are the by far more experienced mathematician, so please don't see it as critique by any means, I only want to understand the other view point. –  Louis Jan 16 at 0:30

2 Answers 2

up vote 1 down vote accepted

The characteristic of an integral domain is the smallest positive integer $n$ such that $n \cdot 1_A= 0_A$. So if $n=1$, then $1_A = 0_A$ in your integral domain $A$. But, we usually (i.e. pretty much always) assume that an integral domain must have unity different from the $0$. Note that a characteristic of an integral domain must always be prime.

share|improve this answer

What does characteristic equal to $1$ mean? It means that $1 x =x = 0$ for all $x.$ So, your $A$ is the zero ring, in particular it has no $1,$ in particular not an integral domain.

share|improve this answer
    
The zero ring certainly does have a $1$, namely $0.\ \ $ –  Bill Dubuque Jan 15 at 22:08

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.