# 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.

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### How Should I show that these $k$-algebras are not Isomorphic?

Question Show that the $k$-algebras $k[x,y]/\langle xy \rangle$ and $k[x,y]/\langle xy-1 \rangle$ are not isomorphic. Attempt At first, I thought $xy=0$. This would mean both $x$ and $y$ are zero ...
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### $\gcd(\text{multiples}(a,b))= \text{lcm}(a, b)$?

I am exploring whether the following assertion holds true in integral domains: $\gcd(\text{multiples}(a,b))= \text{lcm}(a, b)$. Let us make this formal below. Consider two elements $a$ and $b$ in an ...
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1 vote
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### $R = \mathbb R[X,Y]/(XY - 1)$ and $I$ be the ideal of $R$ generated by the image of the element $X - Y$ in $R$. Describe $R/I$

Let $R = \mathbb R[X,Y]/(XY - 1)$ ($\mathbb R$ is the set of real numbers) and I be the ideal of R generated by the image of the element X - Y in R. I want to find a way to describe R/I, i.e. find a ...
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### Proof verification: Is the quotient ring $\Bbb C[x]/(x^2+1)$ an integral domain? [duplicate]

Is the quotient ring $\Bbb C[x]/(x^2+1)$ an integral domain? My solution goes like this: If possible let us assume that $\Bbb C[x]/(x^2+1)$ an integral domain. This means $(x^2+1)$ is a prime ideal in ...
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### Show that if $R$ is an integral domain and $f(x)$ is a unit in the polynomial ring $R[x]$, then $f(x) \in R$ [duplicate]

Show that if $R$ is an integral domain and $f(x)$ is a unit in the polynomial ring $R[x]$, then $f(x) \in R$. Proceed by contraposition. Suppose $R$ is an integral domain, $f(x)$ is a unit in $R[x]$, ...
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1 vote
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### Regarding Vakil's Exercise $10.7.A$

$\newcommand{\O}{\mathscr{O}}\newcommand{\Spec}{\operatorname{Spec}}\newcommand{\Frac}{\operatorname{Frac}}$ In exercise 10.7.A of Vakil's Rising Sea we are tasked with showing that if $A$ is an ...
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### Simplifying Gauss’s Lemma

Hello Math StackExchange Community, I am revisiting Gauss's Lemma in my lecture notes and considering a simplification in its proof. I am proposing to remove the necessity of proving that $\lambda$ is ...
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### $A\otimes_kB$ is an integral domain when $k$ is algebraically closed, and $A$ and $B$ are finitely generated

I am trying to understand this answer in this math stack exchange post I don't see how it works. Let $A$ and $B$ be finitely generated $k$ algebras with $k$ algebraically closed, and suppose that $A$ ...
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### Counterexample of Divisibility of Ideals with Product of Ideals

Given a commutative ring $R$ with unity, we define for $I,J\subseteq R$ ideals $I\ \vert\ J\iff I\supseteq J$ $IJ=\{\sum_i a_ib_i:a_i\in I, b_i\in J\}$ For every commutative unitary ring $R$ it ...
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### Is this the correct negation of the definition and criterias for Irreducible elements in integral domain?

Background: Definition: If $R$ is an integral domain, $p\in R$ is called an irreducible element (and is said to be irreducible in $R$) if it satisfies the following conditions: (1) $p\neq 0$ and $p$ ...
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1 vote
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### ID's, PID's, Noetherian rings and valuation rings: implications amongst them

I am trying to establish some implications between being an ID, a PID, a Noetherian ring and a valuation ring. First of all, I know that PID $\Rightarrow$ Noetherian, because in a PID every ideal is ...
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### What is the meaning of the abbreviation id(x,y)

In the subject of the change of variables via integration of 2-forms on $\mathbb{R}^2$ for a function $\omega=f(x,y) dx \wedge dy$ over $D \subseteq \mathbb{R}^2$, the teacher says that we go from a ...
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### Is kernel of evaluation map on $R[x]$ always principal if $R$ is an integral domain?

Suppose $R$ is an integral domain and $E \supseteq R$ is a ring extension of $R$. Let $\alpha \in E$ commute with all of $R$, and consider the evaluation homomorphism $\phi_\alpha : R[x] \to E$ given ...
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### Prove that if $P$ is a prime ideal of a non-commutative von Neumann regular ring $R$, then $P$ is a maximal ideal

Let $R$ be a ring with identity elements (not necessarily commutative). A ring $R$ is called a von Neumann regular ring if for every $a\in R$ there is $b\in R$ such that $a=aba$. How do I prove that ...
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### If $R$ is a ring such that $\forall x\in R, x^n=x$ for some $n>1$ then when $P$ is prime, why is $R/P$ finite?

I was recently reading a post on MSE which had an argument like: If $P$ is a prime ideal of a ring $R$ all of whose elements satisfy $x^n = x$, then $R/P$ is an integral domain with the same property....
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### Prove $\cap_{n\geq0} I^n = (0)$

Let $A$ be an integral domain and noetherian and let $I \subset A$ a proper ideal such $I*\cap_{n\geq0} I^n = \cap_{n\geq0} I^n$. Prove that $\cap_{n\geq0} I^n = (0)$ I'm trying to prove it by getting ...
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### Free submodules of an integral $R.$

I was told by the author of the answer here Showing that the rank of $M$ is exactly $1.$ that: Free submodules of an integral domain $R$ are exactly the principal ideals of $R.$ I am wondering which ...
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### Every finite integral domain is a field (why is it commutative?)

Okay, a finite integral domain is a finite ring $D$ such that for every $a, b \in D, ab = 0$ iff $a = 0$ or $b = 0$. We want that $D$ is a field, meaning it has a unit element ($1$), every element is ...
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### Number of zero divisors of a ring [duplicate]

I am given the ring $\mathbb{Z}_8 \times \mathbb{Z}_{10}$, and I am asked to find the number of zero divisors. I have counted $23$, not manually, but taking into account several aspects such as what ...
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### Is $I^n \not = I^{n+1}$ for all non-zero proper ideals $I$ of an integral domain? [duplicate]

For $R$ a (commutative with 1) integral domain, is it possible to have $I^n = I^{n+1}$ for some non-zero proper ideal $R$? I realise that in the case of a Noetherian domain, we can apply Krull's to ...
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### Show that $K[x_1,x_2,x_3,x_4] / \langle x_1x_4 - x_2x_3 \rangle$ is an integral domain of dimension $3$

I am stuck at the following exercise from Gathmann's notes on Algebraic Geometry on page 21: Let $R = K[x_1,x_2,x_3,x_4] / \langle x_1x_4 - x_2x_3 \rangle$. Show that $R$ is an integral domain of ...
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### Tensor product of a polynomial ring and the quotient of a polynomial ring modulo a prime ideal

Let $S,R$ be two polynomial rings over a field $K$ in a finite number of variables, and let $I$ be a prime ideal of $R$. Then is it true that the tensor product over $K$ between $S$ and $R/I$ is a ...
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### $A[X, Y ]/(X^2+Y^2−1)$ is integral domain

Deduce that if $A$ is a unique factorization domain (UFD) of char- acteristic zero, then $A[X, Y ]/(X^2+Y^2−1)$ is an integral domain. Let $B = A[X, Y]/(X^2 + Y^2 - 1)$, and suppose that $f(X, Y)$ ...
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### Is an element of a quotient field of $\mathbb{Z}$ a number or a set?

In my book, it is mentioned that the Quotient Field of the Integral Domain of Integers is the Field of Rational Numbers. However, I have a confusion- the Quotient Field of $\mathbb{Z}$ would be the ...
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### Cancellative property in integral domains

I am currently studying integral domains and see that if $ab = ac$ and $a \neq 0$, then we have $b = c$. I was wondering whether the same is true for three nonzero subsets $I, J, K$ of an integral ...
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### How does partial fraction expansion generalize to fractions of integers? Why is it not unique, in that case?

The Wikipedia page for partial fraction expansion mentions that it can be generalised to "regular" fractions, i.e. fractions of integers: https://en.wikipedia.org/wiki/...
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