# Tagged Questions

For questions regarding integral domains, their structures, and properties.

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### $R$ be a Noetherian domain , $t\in R$ be a non-zero , non-unit element , then is it true that $\cap_{n \ge 1} t^nR=\{0\}$?

Let $R$ be a Noetherian domain, $t\in R$ be a non-zero, non-unit element, then is it true that $$\bigcap_{n \ge 1} t^nR=\{0\} \text{?}$$ It almost feels like the nilradical (which is zero for any ...
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### $R$ be an integral domain , $x \in R$ , $I$ an ideal such that $I+\langle x \rangle , (I:x)$ are principal ideals , then is $I$ a principal ideal?

Let $R$ be an integral domain , $x \in R$ , $I$ be an ideal such that $I+\langle x \rangle$ and $(I:x):=\{r \in R : rx \in I\}$ both are principal ideals , then is $I$ also a principal ideal ?
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### If $R$ is an integral domain with unity having only finitely many subdomains (not necessarily with unity), then is $R$ finite?

If $R$ is an integral domain with unity having only finitely many subdomains (not necessarily with unity), then is it true that $R$ is finite ? (I know that there are infinite domains with unity, ...
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### $D$ be a UFD, if an element of $D$ is not a square in $D$ then is it true that, that element is not a square in the fraction field of $D$?

Let $D$ be a UFD, let $F$ be the field of fractions of $D$, let $a \in D$ be such that $x^2 \ne a, \forall x \in D$. Then is it true that $x^2\ne a ,\forall x \in F$ ? (This problem is motivated ...
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### Polynomial ring, ideals and Spec

Morning everyone, I want some hint about this. i) Determine all ideals of $\frac{\Bbb{R[X]}}{<X^3-1>}$ where $R$ is real set ii)Is $\frac{R[X]}{<X^3-1>}$ integral Domain iii)...
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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 ...
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### Integral closure and $\bigcap \mathfrak{a}^n$

Let $R$ be a domain such that $\bigcap_{n=1}^\infty \mathfrak{a}^n=0$ holds for all proper ideals $\mathfrak{a}$ of $R$ (this holds, for example, if $R$ is Noetherian). Let $K$ be the quotient field ...
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### Meaning of $Z\oplus Z$

I am a beginner in Ring Theory and just started Integral Domains. In my textbook, the following was stated : $Z\oplus Z$ is not an integral domain. I can't understand this. I know $\oplus$ ...
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### Can we characterize all infinite Euclidean-domains having exactly one invertible element?

$\mathbb Z_2$ and $\mathbb Z_2[x]$ are two euclidean-domains having exactly one invertible element ; my question is ; Can we characterize all euclidean domains $D$ having exactly one invertible ...
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### Prove that $R = K\langle x,y,z\rangle/\langle x^2 - yz\rangle$ is an integral domain [closed]

Let $R = K\langle x,y,z\rangle/\langle x^2 - yz\rangle$ be an analytic algebra. I am trying to prove that $R$ is an integral domain. Basically I know that if $\langle x^2 - yz\rangle$ is a prime ...
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### If $R$ is an integral domain and $R[x]$ is an euclidean domain, then $R$ is a field [closed]

Is this obvious? I cannot see that this is true. The converse is fairly obvious though. I tried to show $(x)$ is a maximal ideal and try the quotient but failed. I will appreciate any help.
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### $\gcd({p_1}^{n_1},{p_2}^{n_2})$ associates with $1$ if $p_1,p_2$ are prime and do not associate

$R$ is a integral domain, $p_1$ and $p_2$ are prime, $p_1$ and $p_2$ do not associate, $n_1,n_2 \ge 1,n_1,n_2 \in \mathbb N$, I need to show that $g:=\gcd({p_1}^{n_1},{p_2}^{n_2})$ associates with $1$...
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### Relationship between modules and maximal ideals of a commutative ring [closed]

Let $A$ be an integral domain, $M$ an $A$-module, and $m\in M$. Now for all maximal ideals $\mathfrak{m}$ there exists an $n\notin \mathfrak{m}$ such that $nm=0$. Why does this mean that $m=0$?
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### Extension of the fields of fractions of integral domains

Let $A$ and $B$ be integral domains, and let $\varphi:A\to B$ a ring homomorphism. We can give $B$ a structure of $A$-module by saying $ab = \varphi(a)b$. Suppose that $B$ is a finitely generated $A$-...
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### Determine the integral closure of a ring.

Let $R=F[X,Y]/(Y^2-X^3)$. Determine the integral closure of $R$ in its quotient field. I guess I should reduce the problem to some statement related to $F[X]$. For $F$ of characteristic not equal to ...
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### Definition of an Integral Domain in the second edition of Herstein's Topics in Algebra

I think a definition in Herstein's Topics in Algebra needs to be modified and I am asking this question to make sure I am not missing something. Herstein defines an integral domain as a commutative ...
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### Let $D$ be a principal ideal domain. Show that every proper ideal of $D$ is contained in a maximal ideal of $D$.

I know that a PID must satisfy the Ascending chain condition. So Im guessing its going to involve that in the argument some way but Im not sure how to prove it.
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### Example of finite field extension where root not separable

My class notes has as theorem (without proof): "Let $K/F$ be finite field extension, with $K=F(\alpha_1,\ldots,\alpha_n)$ and $\alpha_k$ is separable for all $k$. Then $K/F$ is separable". My ...
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### Existence of GCD in UFD

I proved that any two elements in PID have GCD and it can be expressed as linear combination of those two elements. I know that even in case of UFD GCD exists but it may not be expressed as linear ...
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### In an Integral Domain is it true that $\gcd(ac,ab) = a\gcd(c,b)$?

In my algebra class I was given as homework assignment to prove that: Given an integral domain $A$ and $a,b,c,d,e \in A$ then if $d = \gcd(b,c)$ and $e = \gcd(ac, ab)$ then $e = ad$. It is easy ...
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### Show $\mathbb {R}[x,y]/(y^2-x, y-x)$ is not an integral domain

Let $\mathbb{R}[x,y]$ denote the polynomial ring in two variables $x$, $y$ over $\mathbb{R}$, and let $I = (y^2-x,y-x)$ be the ideal generated by $y^2-x$ and $y=x$. Show that $$\mathbb{R}[x,y]/I$$ is ...
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### Give an example of a homomorphism from an integral domain $R$ to a field which does not factor through the inclusion $R\to\mathrm{Frac}(R)$.

All homomorphisms from fields to fields factor through this inclusion. Thus, the integral domain must not be finite. This is all I have gotten so far. Best regards.
### $R$ be a commutative ring with unity such that every prime ideal contains no non-zero zero divisor , then is $R$ an integral domain?
Let $R$ be a commutative ring with unity such that every prime ideal contains no non-zero zero divisor (i.e. if $P$ is a prime ideal and $x,y \in P$ with $xy=0$ then either $x=0$ , or $y=0$). Then is \$...