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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Characterization of right noetherian rings

Here's a quick question on noetherian rings. I know that for a ring $R$, the following are equivalent. $R$ is left noetherian Every finitely generated left $R$-module is noetherian Every submodule ...
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1answer
53 views

A ring with finite dimensional vector space structure is noetherian?

Let $K$ be a field and $R$ a ring with finite dimensional vector space structure over $K$. Is $R$ necessarily a Noetherian ring? If $K \subset R$, then any ideal in $R$ is also a subspace and, since ...
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1answer
33 views

A two sided ideal of a Noetherian ring

$R$ is a left Noetherian ring with a minimal left ideal. Consider the set of minimal left ideals of $R$ ordered by inclusion. Then there is a maximal element $\mathfrak b= \bigoplus_{i\in I} \mathfrak ...
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1answer
31 views

Graded ring, and its homogeneous ideals : $ I = \displaystyle \bigoplus_ {n \in \mathbb {Z}} (I \bigcap B_n) $

Let $ B = \displaystyle \bigoplus_{n \in \mathbb {Z}} B_n $ be a graded ring. Let $ I $ be an ideal of $ B $. Why is $ I = \displaystyle \bigoplus_ {n \in \mathbb {Z}} (I \bigcap B_n) $ equivalent to ...
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3answers
34 views

Example of ring $R$ with ideals $I\neq J$ such that $R/I \cong R/J$ as modules

It's easy to prove that if $I$, $J$ are two-sided ideals and $R/I\cong R/J$ as modules over $R$, then $I=J$. What about left ideals? Is there a simple counterexample? I believe I've found an answer, ...
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2answers
106 views

A question about algebraically closed fields

A field $\mathbb{K}$ is said to be algebraically closed in practice if every polynomial over $\mathbb{K}$ of positive degree less than or equal to $10^{10}$ has zero belonging $\mathbb{K}$. The ...
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3answers
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Finitely generated ideal in Boolean ring; how do we motivate the generator?

This problem is Exercise 11.3 in Atiyah/Macdonald Commutative Algebra. They ask to prove every finitely generated ideal in a Boolean ring is in fact a principal ideal. The question has been answered ...
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1answer
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My question is about the definition of a map called the “reduction map”.

Let $G$ be a group and $N$ normal in $G$. I have read about a map $\alpha : G\rightarrow \frac{G}{N}$ called the reduction map mod $N$. I would love if someone could please explain this to me. Is it ...
3
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2answers
119 views

If for any two principal ideals one contains another, then for any two ideals one ideal contains another

Let $R$ be a commutative ring with identity. Assume that for any two principal ideals $Ra$ and $Rb$ we have either $Ra\subseteq Rb$ or $Rb\subseteq Ra$. Show that for any two ideals $I$ and $J$ in ...
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1answer
20 views

ideal,ring,flat module,modules over R

Is there a characterization of modules (AND equivalent characterizations of rings R) over integral domains R with the property that each left ideal in R is flat?When all left ideals are ...
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48 views

A Question about the Proof of Eisenstein's Irreducibity Criterion

Statement: Let $f(x) = a_n x^n + a_{n-1} x^{n-1}+ \cdots + a_0 \in \mathbb Z[x]$. If there is a prime $p$ such that $p \nmid a_n, p \mid a_{n-1}, \dots,p \mid a_0$ and $p^2 \nmid a_0 $, then $f(x)$ ...
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0answers
37 views

When does $n$-dimensional algebra have $m$-dimensional faithful representation?

Suppose we have an $n$-dimensional associative unital algebra $A$ over a field $k$ (assume $\operatorname{char}(k)=0$ and maybe even $k$ is closed). I would like to know what is the minimal ...
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1answer
19 views

Embedding the base ring in the augmentation ideal of a group algebra

Let $G$ be a finite group. Then the group algebra $\mathbb{Q}G$ trivially contains $\mathbb{Q}$. But when (i.e. for which $G$) does the augmentation ideal $I_G=\{\sum_{g\in G} r_g\,g \mid \sum_{g\in ...
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2answers
26 views

finite boolean ring order is $2^n$

let $R$ be a finite boolean ring. prove that $|R|=2^n$ for some $n\in\mathbb N$. I know that $R$ is commutative and for every element $a\in R\space a+a=0$ and $a^2=a$
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1answer
35 views

Irreducibility of a polynomial over a field

I'm trying to show that the polynomial $f(x) = \frac{x^5}{32}-3x-2$ is irreducible over $\mathbb Q$. Obviously $f$ doesn't have a root over $\mathbb Q$ so I tried to use Gauss lemma for $32f$ and ...
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Proving $R=\{a+b\sqrt3\mid a,b \in \Bbb Z\}$ is Euclidean. [on hold]

Let $R=\{a+b\sqrt3\mid a,b \in \Bbb Z\}$ A. Prove $R$ is a Euclidian domain with respct to the norm $N(a+b\sqrt3)= |a^2-3b^2|$. B. Divide $1+\sqrt3$ by $2+\sqrt3$ in $R$.
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2answers
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Is $2x^2+4$ reducible over $\mathbb C$?

I am not sure if I making some very fundamental mistake. But Gallian says that $2x^2+4$ is reducible over $\mathbb C$. If $D$ be an integral domain. A polynomial $f(x)$ from $D[x]$ is said to be ...
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0answers
54 views

Problems from Polynomial Rings. My attempt shown.

$1.$ Let $f(x) =a_mx^m+a_{m-1}x^{m-1}+ \cdots +a_0$ and $g(x) = b_nx^n+b_{n-1}x^{n-1}+ \cdots +b_0$ belong to $\mathbb Q[x]$ and suppose that $f \circ g \in \mathbb Z[x].$ Prove that $a_ib_j$ is an ...
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0answers
29 views

Text on Witt vectors that are accessible to undergraduate students

I am looking for a thorough text on Witt vectors that is accessible to an undergraduate student that have completed the following courses: Calc 1, 2, Linear Algebra and Abstract Algebra. (In Norway, ...
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35 views

Simple Cogenerator for the category of left $R$-modules

I want to prove that if the category of left $R$-modules has a simple cogenerator $M$, then $R$ is a simple ring. (Here, $R$ is an arbitrary ring with $1$.) My try is to take an ideal $I$ of $R$ ...
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1answer
55 views

If $F$ be a field, then $F[x]$ is a principal ideal domain. Does $F$ have to be necessarily a field?

If $F$ be a field, then $F[x]$ is a principal ideal domain. Does $F$ have to be necessarily a field? My Thoughts: Suppose instead of $F$, we take the set of polynomials $R[x]$ over a commutative ring ...
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2answers
36 views

A doubt about lower nil radical while proving 2-primality of ring.( Baer-McCoy Radical)

I was proving that a reversible ring is 2-Primal for an exercise in T.Y Lam's book, but I got stuck. Here is where I'm stuck: let $a$ be a nilpotent element of $R$ with $a^n=0$. Then using ...
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1answer
33 views

Global dimension regular rings of finite type

Have I made an error in my reasoning? If $k$ is a field, $A$ is a commutative regular $k$-algebra of finite type and ${\mathfrak{m}}$ is a maximal ideal in $A$ then since $Ext_{A_{\mathfrak{m}} ...
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2answers
38 views

A ring with prime characteristic

Let $p$ be a prime and let $R$ be a commutative ring with characteristic $p$. Prove that the number of elements of the set $$S_k=\{x\in R\;\lvert \;x^p=k\}\quad \text{for} \quad k\in ...
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0answers
43 views

Is every local ring the localization of some other ring?

One way of constructing a local ring is to start with any commutative ring, and localize all the elements outside of some maximal ideal (i.e., adjoining inverses to all those elements). But I'm ...
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2answers
42 views

Prove that a polynomial of degree $n$ over a commutative ring with zero divisors may have more than $n$ zeroes?

A polynomial of degree $n$ over a commutative ring with zero divisors may have more than $n$ zeroes. Attempt: Let $R$ be the commutatve ring which has a zero divisor $a \neq 0$. Then $\exists~~b \in ...
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41 views

prove that if $p(x)\in R[x]$ is reducible over $F[x]$ then $p(x)$ is reducible over $R[x]$.

let $R$ be a unique factorization domain and let $F$ be its field of fractions. Prove that if $p(x)\in R[x]$ is reducible over $F[x]$ then $p(x)$ is reducible over $R[x]$.
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1answer
39 views

Show that if M, N are non-zero commutative rings, then M×N always has zero divisors, and is not an integral domain or a field.

Show that if M, N are non-zero commutative rings, then M×N always has zero divisors, and is not an integral domain or a field. How do I do this?!
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1answer
63 views

Irreducibility of some multivariate polynomials

Consider the polynomials $xw-yz\in A[x,y,z,w]$ and $x^n+y^n+z^n\in A[x,y,z]$, where $A$ is a commutative ring. I am curious to know what conditions on $A$ (factorial ring, algebraically closed field, ...
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2answers
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Prove that the field of quotients of an integral domain $D$ is the smallest field containing $D$. . My Attempt Shown

Let $D$ be an integral domain and let $F$ be the field of quotients of $D$. Show that if $E$ is any field that contains $D$, then, $E$ contains a sub field that is ring isomorphic to $F$. Hence, the ...
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1answer
26 views

Suppose that $n$ divides $m$ and that $a$ is an idempotent of $Z_n$. Show that the mapping $x \rightarrow ax$ is a ring homomorphism

Suppose that $n$ divides $m$ and that $a$ is an idempotent of $Z_n$. Show that the mapping $x \rightarrow ax$ is a ring homomorphism Attempt: Let $\Phi: Z_m \rightarrow Z_n$ be a ring homomorphism ...
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2answers
34 views

Let $F$ be a field. Show that the field of quotients of $F$ is ring isomorphic to $F$. My Attempt Shown

Let $F$ be a field. Show that the field of quotients of $F$ is ring isomorphic to $F$. Attempt: Let $F'$ be the field of Quotients of the field $F$. Let $\Phi:F \rightarrow F'$ such that ...
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1answer
34 views

Finitely generated ring.

Let $R$ be a finitely generated ring then is it true that $\hspace{0.1cm}$$R^2$ $\hspace{0.09cm}$ also finitely generated? My Attempt: I do not find a counterexample. I think it is true. Please ...
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1answer
69 views

What is necessary and/or sufficient for polynomials to provide isomorphic quotientrings?

Let $R$ be a commutative ring (with identity). Let $f,g\in R\left[x\right]$ both be monic polynomials of degree $d$. Then the underlying abelian groups of the rings ...
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2answers
42 views

$R^{(I)} \cong K \oplus H$ where $R^{(I)}$ is free but $K$ is not free

Let $R$ be a commutative ring with unit. Is there an example of a direct sum of $R$-modules $$R^{(I)} \cong K \oplus H$$ where $R^{(I)}$ is free but $K$ is not free ? Clearly $R$ can't be a PID.
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Classification of separable algebras over a commutative ring

A separable algebra over a field can be classified as a finite product of matrix algebras over division algebras whose centers are finite dimensional separable field extensions of the field. (See ...
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1answer
19 views

an ideal of matrix ring which is projective

Let $K$ be a field and $$ A=\left\{ \begin{pmatrix} a&b&c\\ d&e&f\\ 0&0&g \end{pmatrix} :a,\dots,g\in K \right\}, $$ then $$ J=\left\{ \begin{pmatrix} 0&0&c\\ ...
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40 views

The $i^{th}$ prime in a given ring R

When I say that $p_1=2$, I mean that the first prime in the standard ring of integers $(\mathbb{Z},*,+)$ is $2$. I was wondering whether the notion of ordering the primes like this can be generalized ...
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1answer
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Kahler forms of a smooth affine algebra vanish eventually?

If $k$ is a Noetherian ring, then do the Kahler forms of a smooth affine $k$-algebra of dimension $d$ vanish above $d$? I mean is: $\Omega^{d+1}_{A|k}\cong 0$?
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1answer
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$z\in\mathfrak R$ iff for every $a\in A$ there is $w$ for which $z+w=zaw=waz$.

In his BAII, Jacobson gives the following exercise, which he attributes to McCrimmon. Show that $z\in\mathfrak R(A)$ iff for each $a\in A$ there exist $w\in A$ such that $z+w=zaw=waz$. I have ...
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Units and Primes in a Ring

Is it true that units in a ring (maybe involves in quaternions) have norm of 1? (norm of 1 does not imply that it is a unit, right?) What about the statement that the number is prime if and only if it ...
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1answer
146 views

Direct product of finitely many Noetherian non-unital rings is Noetherian

Let $A_1, A_2,...,A_n$ be Noetherian rings (not necessarily unital). Is the direct product $A:=A_1×A_2×⋯×A_n$ necessarily a Noetherian ring? If $A_1, A_2,...,A_n$ are unital, then one can prove ...
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108 views

A field with characteristic $0$ contains $\mathbb Q$

To prove that a field $F$ with characteristic $0$ contains $\mathbb Q$, the following lemma is used. Lemma: Let $R$ b a ring with unity. If the characteristic of $R$ is $0$, then $R$ contains a ...
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What is $\operatorname{Ass}\operatorname{Ext}^i(M,N)$?

This is exercise 1.2.27 of Bruns-Herzog: Let $R$ be a Noetherian ring, $M$ a finite $R$-module and $N$ an arbitrary $R$-module. Deduce that $\operatorname{Ass}(\operatorname{Hom}_R(M,N)) = ...
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76 views

Number of elements and units in quotients of $\mathbb Z[i]$

I was given this problem but have no idea how to solve it. How many elements are there in $\Bbb Z[i]/(3)$? In $\left(\Bbb Z[i]\right)/(3+2i)$? In $\Bbb Z[i]/(5)$? How many units are there in each of ...
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Why is an extension $\bar \phi : F(x) \rightarrow F(a)$ an isomorphism if $\phi : F[x] \to F(a)$ is injective?

Why is $\bar \phi : F(x) \rightarrow F(a)$ an isomorphism given $\phi : F[x] \rightarrow F(a)$ satisfy $\ker \phi = \{0\}$ ? I've been trying to figure out why $\bar \phi$ is an isomorphism, and ...
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80 views

Suppose that R is a commutative ring with unity such that for each $a$ in $R$ there is an integer $n > 1\mid a^n =a$. Every prime ideal is maximal?

Suppose that R is a commutative ring with unity such that for each $a$ in $R$ there is a positive integer $n > 1$ such that $a^n =a$. Prove that every prime ideal of $R$ is a maximal ideal of R. ...
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1answer
45 views

Localizations of $ \mathbb{Z}_{p^k}$

Let $S \subseteq \mathbb{Z}_{p^k} $ be a multiplicative subset, where $p$ is a prime number, $k$ an integer. Is it true that $$S^{-1} \mathbb{Z}_{p^k} \cong \mathbb{Z} /n\mathbb{Z} $$ for some ...
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1answer
33 views

Congruence in localization of rings

Please help me to prove for all maximal ideals $\mathfrak{m}$ of $R$, $(aR/a^2R)_\mathfrak{m}\cong (aR)_\mathfrak{m}/(a^2R)_\mathfrak{m}\cong aR_\mathfrak{m}/a^2R_\mathfrak{m}$, where $R$ is a ...
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3answers
50 views

In a principal ideal domain, prove that every non trivial prime ideal is a maximal ideal. What could be wrong in this approach?

In a principal ideal domain, prove that every non trivial prime ideal is a maximal ideal Attempt: Let $R$ be the principal ideal domain. A principal ideal domain $R$ is an integral domain in which ...