Questions tagged [integral-domain]

For questions regarding integral domains, their structures, and properties. This tag should probably be accompanied by the Ring Theory tag. This tag is not for use for questions regarding integrals in analysis and calculus.

An integral domain is a commutative ring with identity and no zero divisors. That is, a commutative ring $R$ with $1$ is an integral domain if and only if for all $a, b \in R$,

$$ab = 0 \implies a = 0 \text{ or } b = 0$$

Alternatively, a commutative ring with $1$ in which the ideal $\{0\}$ is prime is an integral domain. Note that some authors do not require an integral domain to have a unit $1$.

The prototypical example of an integral domain is the ring of integers, $\Bbb{Z}$, and all fields are integral domains.

Source: Integral domain on Wikipedia.

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Why is a finite integral domain always field?

This is how I'm approaching it: let $R$ be a finite integral domain and I'm trying to show every element in $R$ has an inverse: let $R-\{0\}=\{x_1,x_2,\ldots,x_k\}$, then as $R$ is closed under multiplication $\prod_{n=1}^k\ x_i=x_j$, therefore…
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Where Fermat's last theorem fails

It's fairly well known that Fermat's last theorem fails in $\mathbb{Z}/p\mathbb{Z}$. Schur discovered this while he was trying to prove the conjecture on $\mathbb{N}$, and the proof is an application of one of his results in Ramsey theory, now known…
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A subring of the field of fractions of a PID is a PID as well.

Let $A$ be a PID and $R$ a ring such that $A\subset R \subset \operatorname{Frac}(A)$, where $\operatorname{Frac}(A)$ denotes the field of fractions of $A$. How to show $R$ is also a PID? Any hints?
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How to prove that the Frobenius endomorphism is surjective?

$R$ is a domain with characteristic $p$ ($p$ is prime). There is a homomorphism $f : R \to R$, $f(a)=a^p$. $f$ is called the Frobenius endomorphism. And I have known this. When $R$ which is mentioned above is also a field, it is said that $f$ is an…
Andylang
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An integral domain $A$ is exactly the intersection of the localisations of $A$ at each maximal ideal

This result appears to be ubiquitous as an algebra exercise. How do you prove this result? Let $A$ be an integral domain with field of fractions $K$, and let $A_{\mathfrak{m}}$ denote the localisation of $A$ at a maximal ideal $\mathfrak{m}$…
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How to show every field is a Euclidean Domain.

I'm having trouble proving this. This is what I have so far: Let $F$ be a field. Let $v(x) \rightarrow 1$ for all $x$ not equal to $0$. So if we let $x$ be in $F$ where $x$ not zero then we can write $x$ as: $x=qy+r$ for some $y$ in $F$. If $r$ not…
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Prove that an infinite ring with finite quotient rings is an integral domain

How can we show that if $R$ is an infinite commutative ring and $R/I$ is finite for every nonzero $I \unlhd R$, then $R$ is an integral domain? I tried proceeding by contradiction: assume $a$,$b$ $\in R \backslash \{0\}$ and $ab=0$; then $R/(a)$…
Harry Macpherson
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A fraction field is not finitely generated over its subdomain

I'm looking for proofs of the following fact. Suppose that $R$ is a domain which is not a field with fraction field $K$. Then $K$ is not finitely generated as $R$-module. I know this fact is true, at least, when $R$ is Noetherian and I guess it is…
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Can we characterize all infinite PID s whose group of units is singleton?

I am looking for a way to characterize all infinite PID s having exactly one unit i.e. invertible element ( finite PID s are not interesting , they are all fields ) . The only such example I know of is $\mathbb Z_2[x]$ . Towards characterizing such…
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Integral domain that is not a factorization domain

I am looking for rings that are integral domains but not factorization domains, that is, rings in which it is not possible to express a nonzero nonunit element as a product of irreducible elements. Do you know any example?
user14174
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Looking for an example of a GCD domain which is not a UFD

I know that every UFD (unique factorization domain) is a GCD domain i.e. g.c.d. of any two elements, not both zero, exists in the domain. I am looking for an example of a GCD domain which is not a UFD. I have not been able to find mainly for the…
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Show that $A[X]/(aX+b)$ is an integral domain

Let $A$ be an integral domain, $a$ and $b \in A-\{0\}$, and let $B = A[X]/(aX+b)$. Show that, if $Aa \cap Ab=Aab$, then $B$ is an integral domain. My attempt at proof (following a hint). Denote by $K$ the field of fractions of $A$. Let $\phi: A[X]…
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Cancellation law in a ring without unity

When discussing rings, integral domains, fields etc, I'm told that the cancellation law holds in any ring that has no zero divisors. By cancellation law, I mean that if we have no zero divisors, we can look at the equation $ab = ac$ and "cancel" the…
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Is an infinite field always isomorphic to a non-trivial fraction field?

Out of the blue, I asked myself the following question: Is an infinite field always isomorphic to the fraction field of an integral domain which is itself not a field? Please note that the above setup avoid answering by a field is its own fraction…
C. Falcon
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Irreducible elements for a commutative ring that is not an integral domain

Why does the definition of an irreducible element require us to be in an integral domain? Why can we not define an irreducible element exactly the same in a commutative ring that is not an integral domain? We have that an element is irreducible…
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