This tag is for questions about rings, which are a type of algebraic structure studied in abstract algebra and algebraic number theory.

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$R$ be an infinite commutative ring with unity , also a PIR , such that for every non-zero ideal $I$ of $R$ , $|I|=|R|$ ; is $R$ an Integral Domain?

Let $R$ be an infinite commutative ring with unity , which is also a PIR , such that for every non-zero ideal $I$ of $R$ , $|I|=|R|$ i.e. the sets $I$ and $R$ are bijective ; then is it true that $R$ ...
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When is the polynomial $\prod\limits_{i=1}^n\,\left(x-a_i\right)+1$ reducible in $\mathbb{Z}[x]$?

This post is inspired by Prove that the polynomial $\prod\limits_{i=1}^n\,\left(x-a_i\right)-1$ is irreducible in $\mathbb{Z}[x]$.. (A) Find all positive integers $n$ and integers $a_1,a_2,\...
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108 views

Smallest Two-Sided Nearring

For those who are unfamilar with nearrings, here is a definition. A two-sided nearring is a triplet $(N,+,\cdot)$, where $(N,+)$ is a group and $(N,\cdot)$ is a semigroup such that we have both ...
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$(f_1(x), f_2(x), …, x-a) = (f_1(a), …, f_r(a), x-a)$ with $a \in R$, $R$ is a commutative ring, $f_i(x) \in R[x]$

Let $R$ be a commutative ring, $a \in R$, and $\forall i = 1, ...,r \ \ f_i(x) \in R[x]$. Prove the equality of ideals $(f_1(x), ..., f_r(x), x-a ) = (f_1(a), ...f_r(a), x-a)$. That is, $\forall ...
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1answer
52 views

Build a reduced ring starting from an ordinary one

This may be easier than I think, but still I can't seem to wrap my head around it. I've learnt that if we take a ring $R$ and quotient it for a (two-sided) ideal $I \subset R$ which is radical, the ...
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2answers
38 views

Given a ring $R$ and a ring extension $R'$, if $r=r's$ where $r'\in R'\setminus R$, does that mean that $s\not\mid r$ in $R$?

Given a ring $R$ and a ring extension $R'$, if $r=r's$ where $r'\in R'\setminus R$ and $r,s\in R\setminus\{0\}$, does that mean that $s\not\mid r$ in $R$? I was thinking for example in $\Bbb{Z}$, ...
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1answer
40 views

An Equality in Ring Theory

Let $R$ be a ring with identity (commutative or not), and assume $J(R)$ to be its Jacobson radical. Let $e\in R $ be an idempotent and $x\in J(R)$ be a given fixed element. I am searching for ...
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2answers
20 views

For each ideal $I$ of $B$, there seems to be a corresponding morphism $f/I :A/f^{-1}I \rightarrow B/I$. Is this right?

(All my rings are commutative with $1$.) Suppose $f : A \rightarrow B$ is a morphism of rings. Then for each ideal $I$ of $B$, there seems to be a corresponding morphism $$f/I :A/f^{-1}I \rightarrow ...
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3answers
52 views

Form of the elements of a localization

If I have a ring $R$, a multiplicatively closed subset $U\subset R$, and consider an element of the localization: $\frac{r}{r'} \in U^{-1}R$, can I then assume without loss of generality that $r'\in U$...
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0answers
17 views

Radical of a ring of size p^7

Suppose $p$ is prime, $R$ is a ring of size $p^7$, and $J$ is the Jacobson Radical of $R$. In general, $R/J$ is isomorphic to a direct product of matrix rings over fields. Does there exist a ring of ...
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2answers
56 views

Let $\mathfrak{p},\mathfrak{q}$ be prime ideals of the ring $R$ s.t. $\mathfrak{p}+\mathfrak{q}\neq 1$. Must $\mathfrak{p}+\mathfrak{q}$ be prime?

A previous question asked this in full generality, without the condition that $\mathfrak{p}+\mathfrak{q}\neq 1$; however, I was wondering if the statement is false only in that particular edge case, ...
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1answer
437 views

Is this matrix decomposition possible?

Given a $2\times2$ matrix $S$ with entries in $\mathbb{Z}$ or $\mathbb{Q}$ , when is it possible to write $S=\frac{1}{3}(ABC+CAB+BCA)$ such that $A+B+C=0$, where $A, B, C$ are matrices over the same ...
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25 views

Definition of Irreducible polynomial in terms of the unit of Integral domain.

Let $D$ be an integral domain. A polynomial $f(x)$ from $D[x]$ that is neither the zero polynomial nor a unit in $D[x]$ is said to be irreducible over $D$ if, whenever $f(x)$ is expressed as a product ...
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1answer
48 views

Equality of polynomial functions modulo n

Fix positive integers $m$ and $n$. For all polynomial functions $f,g: \mathbb{Z}^m \to \mathbb{Z}$ define the equivalence relation $\sim$ by $$f \sim g \iff \forall x \in \mathbb{Z}^m \ ( \ f(x) \...
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1answer
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Is the number of isomorphism classes of quotients of a finite dimensional commutative ring over a field finite?

If $A$ is a finite dimensional unital and commutative algebra over some infinite field $k$, what is the number of isomophism classes of rings of the form $A/I$ where $I$ is a proper ideal of $A$? Is ...
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1answer
36 views

Let $R$ be an infinite comutative ring with unity, $M,N$ be $R$-modules, $f:M \to N$ be a surjective module homomorphism; then $|M|=|N ||\ker f|$?

Let $R$ be an infinite commutative ring with unity, $M,N$ be modules over $R$, let $f:M \to N$ be a surjective module homomorphism; then is it true that $|M|=|N || \ker f|$ ($M,N$ are not ...
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1answer
34 views

Do quotients by an ideal carry over a ring isomorphism?

Say we have ring $A$ such that $A \cong B/I$ for some ideal $I$ of $B$, and suppose also that $B \cong C$ for some other ring $C$. Does there exist an ideal $J$ of $C$ such that $A \cong C/J$? Also, ...
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1answer
41 views

Ideals of Unique Factorization Domain

Let R be a commutative ring with unity such that R[x] is UFD. The ideal (x) of R[x] is denoted by I. Then pick the correct statements from below: 1. I is prime. 2. If I is maximal then R[x] is a PID. ...
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1answer
73 views

Is the notion of Stabilizer of a subset A of a group G absurd?

Is the notion of Stabilizer of a subset,A of a group G is absurd? I don't know whether this makes sense or not,but for curiosity i want to know view of experts. UPDATE i'm dealing with ...
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3answers
29 views

Example of a communtative ring with two operations where the identity elements are not distinct?

I was introduced to the definition of a field today, as a communtative ring with two operations, like $\Bbb{R} = \langle R, +, -, \cdot, ^{-1} \rangle$ and all the usual axioms; commutativity, ...
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1answer
44 views

Showing that $c_{i}\equiv 0\pmod{p}$

Let the numbers $c_{i}$ be defined by the power series identity $$\frac{1+x+x^{2}+\ldots+x^{p-1}}{(1-x)^{p-1}}= 1+c_{1}x+c_{2}x^{2}+\ldots$$ Show that $c_{i}\equiv 0\pmod{p}$ for all $i\geq 1$. $\...
3
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1answer
70 views

On graded Artinian Gorenstein algebras

Let $k$ be a field and $R$ an $\mathbb{N}$-graded $k$-algebra that is graded-commutative. Assume that $\dim_k R<\infty$ and that $R$ is Gorenstein (i.e. the injective dimension of $R$ over itself ...
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3answers
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Does integration by parts with “deja vu” have a name?

In some integration by parts problems, such as evaluating the integral of $e^x \cos x$ or $\sec^ 3 x$, one performs integration by parts (possibly more than once, and possibly together with algebraic ...
3
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1answer
28 views

Can we construct a ring of order $15$ without identity not isomorphic to $\mathbb{15}$?

I've proved any ring of order $15$ with identity is isomorphic to $\mathbb{Z}_{15}$ but what if the ring is of order $15$ with no identity element ? Can we construct a ring of order $15$ without ...
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Find an $R$-module homomorphism $f:R\longrightarrow M$ such that $r_0m=0$ implies $r_0f^{-1}(m)=0$ , $(m\in M)$

Let $R$ be a commutative ring with $r_0\in R$ a fixed element, and $M$ be an $R$-module. I search for an $R$-module homomorphism $f:R\longrightarrow M$ such that $r_0m=0$ implies $r_0f^{-1}(m)=0$ , $(...
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0answers
34 views

Idempotents in commutative ring of characteristic 2 form a subring

Question: In a commutative ring of characteristic $2$, want to show that the idempotents form a subring. Subring Test is probably the way to go. It is easy to verify the identity element in ...
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1answer
165 views

How do I prove that $\mathbb{Z}[\sqrt{19}]$ is not a UFD?

How do I prove that $\mathbb{Z}[\sqrt{19}]$ is not a UFD? Moreover, how do I prove that $(7,3+\sqrt{19})$ is not a principal ideal? This is the first time I'm dealing with a quadratic integer ring ...
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1answer
111 views

When are all ring homomorphisms also algebra homomorphisms?

Let $k$ be an algebraically closed field, and let $A,B$ be two unitary $k$-algebras. In general, there are more ring homomorphisms $A\to B$ than there are $k$-algebra homomorphisms. More precisely, ...
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47 views

Commutativity theorem in the ring theorem

Suppose that $R$ is a ring such that for any $x\in R$ there exists $1<n(x)\in \mathbb{N}$ such that $x^{n(x)}-x\in Z(R)$. Prove that $R$ is commutative or if it is not commutative, then the ideal ...
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2answers
58 views

Finite commutative ring with unity and without nilpotent elements

Let $R$ be a commutative ring with unity such that for each $x \in R$ there exists a $n \in \mathbb{N}$, $n>1$, such that $x^n = x$. Then show that $$ R\simeq F_{1}\times F_{2}\times \cdots\times ...
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3answers
53 views

$D$ 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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1answer
27 views

Let $R$ be a commutative ring. For $a \in R$ consider the set $(a) = \{r*a | r\in R\}$. Show that $(a) = R$ if a is a unit of $R$ [duplicate]

I tried some values and I think I got the idea. R is the set of values used in the ring. If I use $\mathbb{Z}$, the units are $\{-1,1\}$. If I take 1 for example, I could use it to get every value in $...
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1answer
42 views

Prime elements with the same norm in a Euclidean domain [on hold]

Does anybody know whether two prime elements with the same norm in a Euclidean domain are necessarily associated? Any help will be very welcome. UPD 1: It was shown that $2\pm i$ are both primes ...
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38 views

Vector space complement to a multiplicatively closed subspace is an ideal

Let $V$ be a vector space over $\mathbb{C}$ of any dimension and suppose we have an associative multiplication $V \times V \to V$ making $V$ into a commutative ring with unity. Let $V=U \oplus W$ be a ...
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1answer
330 views

When is $\mathbb{Z}[\sqrt{d}]$ not an UFD (for $d>1$)?

I was wondering if there is a classification for this: For which $d$ is $D=\mathbb{Z}[\sqrt{d}]$ a UFD, with $d > 1$? For $d \equiv 1 $ (mod $ 4$), $D$ is not a UFD (proof here).
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1answer
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Implication in local noetherian domain of Krull's dimension 1. [duplicate]

Let $(A,m)$ be a local noetherian domain with Krull dimension $1$. Let $k$ be the field $A/m$. I'm trying to prove that if $m/m^2$ has dimension $1$ as a $k$-vector space, then every ideal $I$ ...
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1answer
37 views

Let $M_1$, $M_2$ be Artinian modules over $R$. Then $M_1\times M_2$ is Artinian.

Using exact sequences, it's fairly easy to prove the converse, but I can't figure out how to prove this statement. Suppose we have a descending chain $N_1\supset N_2\supset\cdots$ of $R$-submodules ...
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2answers
39 views

Generalization of a Result on Modular Inverses

Yesterday, I attempted to solve the general system of linear congruences (I'm not sure why I've never tried this before.) \begin{align*} x &\equiv a \pmod{A} \\ x &\equiv b \pmod{B}.\end{...
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4answers
368 views

If two elements in a ED have the same Euclidean norm, are they associates?

Is it obvious that in a Euclidean Domain two elements $x$ and $y$ having the same Euclidean norm are associates? Can someone give me a proof of this?
3
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1answer
30 views

$p(x) \in \mathbb R[x]$ be a polynomial of odd degree , $n>1$ be an integer , then is the function $A \to p(A)$ surjective on $M(n,\mathbb R)$?

Let $p(x) \in \mathbb R[x]$ be a polynomial of odd degree , $n>1$ be an integer , then is the function $f: M(n,\mathbb R) \to M(n, \mathbb R)$ defined as $f(A)=p(A) , \forall A \in M(n,\mathbb R)$...
3
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1answer
40 views

$p(x) \in \mathbb R[x]$ be non-constant polynomial , $n>1$ , the function $A \to p(A)$ is surjective on $M(n, \mathbb C)$?

Let $p(x) \in \mathbb R[x]$ be a non-constant polynomial and $n>1$ , then is it true that the function $f:M(n,\mathbb C) \to M(n, \mathbb C)$ defined as $f(A)=p(A) , \forall A \in M(n, \mathbb C)...
2
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1answer
53 views

what kind of integral domain do the non-infinite surreals form?

https://en.wikipedia.org/wiki/Integral_domain mentions the following chain of inclusions: Principal Ideal domains $\subset$ Unique Factorization domains $\subset$ GCD domains $\subset$ Integrally ...
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1answer
29 views

quick question on ascending chain condition for rings

I know that if $R$ is a commutative ring with an identity in which every ideal if finitely generated then it satisfies the ascending chain condition. Just wondering if the converse is also true?
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116 views

UFD yields height of certain primes at most $1$

Let $R$ be a unique factorization domain. If $P$ is a prime ideal minimal over a principal ideal, is it true that height of $P$ is at most $1$? In case $R$ is Noetherian the result follows due to ...
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3answers
99 views

Given a UFD and two polynomials not equal to zero, product of gcd is a unit

Here is a question from an old exam: 1. a) Let $R$ be a UFD and let $A=a_{0}x^{m}+...+a_{m}\ne 0 , B=b_{0}x^{n}+...+b_{n} \ne 0$ in $R[x]$ with $\gcd(a_{0},...,a_{m})\in R^{*}$, $\gcd(b_{0},...,...
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1answer
49 views

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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0answers
23 views

Factoring polynomials when roots are external to the ring

I shall avoid maths script since I'm typing on a mobile, anyway I think I can do without. I have a question about factoring polynomials over a ring. Let's call R the ring in question. It is clear to ...
1
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1answer
68 views

Universal property, localization of rings and modules, and initial element in a category

Sorry for the confusing title. I just started learning category theory and am very confused about the concept "universal property". I am not even sure whether my "proof" is a proof or is just a ...
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1answer
57 views

Delooping of a ring?

I'm no expert on category theory, so the definition of delooping in the nlab article is a bit over my head. However, I do understand the practical idea that we can think of a group $G$ as a one-object ...
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2answers
37 views

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)...