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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3
votes
1answer
45 views

Zero-square algebra

Let $k$ be a field and let $A$ be a non-unital, commutative algebra over $k$. Suppose that the only ideals of $A$ are $A$ itself and $\{0\}$. The task is to show that $A$ is isomorphic to the ...
0
votes
2answers
23 views

A question about commutative $R$-algebras.

Aluffi describes a commutative $R$-algebra as a ring homomorphism $\alpha:R\to S$ where $S$ is commutative. Going by this definition, how is $R[x_1,x_2,\dots,x_n]$ a commutative $R$-algebra? What is ...
3
votes
2answers
50 views

Let $R$ be a ring . Then prove that $(R/ \langle x \rangle )/ \langle y \rangle \approx R/ \langle x,y \rangle $

Let $R$ be a ring and $x,y \in R$ s.t. $ \langle x \rangle , \langle y \rangle $ are ideals of $R$, then prove that $(R/ \langle x \rangle )/ \langle y \rangle \approx R/ \langle x,y \rangle $ Not ...
1
vote
1answer
50 views

Injectivity of simple modules

If $R$ is a commutative ring with $1$ having a maximal ideal $m$ such that the local ring $R_m$ is a field, how could one check that $R/m$ is an injective $R$-module? If we want to use Baer Lemma, we ...
6
votes
1answer
70 views

Quotients of topological rings

Let $\varphi\colon R\to S$ be a surjective ring homomorphism and let $R$ be a topological ring. Is there some nice characterization of the finest topology on $S$ for with both $S$ becomes a ...
2
votes
1answer
41 views

Prove that for $P(X) \in \mathbb{Z}[X]$ the set $S = \left\{p : \text{prime and }p \mid P(n) \text{ for } n \in \mathbb{Z}^+\right\}$ is infinite

Prove that for $P(X) \in \mathbb{Z}[X]$, $P(x)$ non-constant, the set $S = \left\{p : \text{prime and }p \mid P(n) \text{ for some } n \in \mathbb{Z}^+\right\}$ is infinite. could someone please give ...
2
votes
2answers
79 views

Finding a maximal ideal and a prime ideal in $\mathbb Z_8[x]$

$1.$ Find a maximal ideal and a prime ideal in $\mathbb Z_8[x]$ Attempt: Finding a maximal ideal, I am not sure how do I go about it. $\mathbb Z_8[x]$ is not a $PID$, so there's no use finding ...
1
vote
1answer
39 views

Maximal ideals of a subring

What are the maximal ideals of $R_5=\{x\in\mathbb{Q}|x=a/b,5\nmid{b},a,b\in\mathbb{Z} \}\subset\mathbb{Q}$? I am thinking that they are of the form $R_5(x)$ with x$\in\mathbb{Z}$ since if $I$ a ...
0
votes
1answer
50 views

Multiplicative homomorphism of Euclidean domains: Do irreducibles never map to reducibles?

Suppose I have a multiplicative homomorphism $\phi: R \to S$ between two euclidean domains. Is it true that an irreducible element of $R$ never maps to a reducible element of $S$? Edit, after first ...
2
votes
2answers
76 views

When a finite local ring $R$ has $-1$ as a square in $R^\times$?

Let $R$ be a finite local ring with maximal ideal $M$ such that $|R|/|M|\equiv 1\pmod{4}$. Then $-1$ is a square in $R^\times$ (that is, there exists $u\in R^\times$ such that $u^2=-1$) if and only ...
1
vote
0answers
34 views

Almost Dedekind (not Noetherian) [duplicate]

A domain $R$ is a Dedekind domain iff it is integrally closed, Noetherian with Krull dimension at most one. I think they put Noetherian in the definition not only to make proofs easier but also ...
4
votes
2answers
30 views

Show that an integral domain with every strictly decreasing chain of ideals $I_1 \supset I_2\supset \cdots $ finite in length is a field. [duplicate]

Show that an integral domain with the property that every strictly decreasing chain of ideals $I_1 \supset I_2\supset \cdots $ must be finite in length is a field. Attempt: We need to show that every ...
0
votes
1answer
62 views

Rank of a module when the base ring is not a domain

Suppose $R$ is a commutative Noetherian local ring with $1$, which is not a domain. Let $M$ be a (non-free) finite $R$-module. What is meant by rank of $M$ in this case?
0
votes
2answers
71 views

Graded rings and modules - which book to refer to?

Could you recommend a book with a good treatment of the theory of graded rings and modules?
0
votes
0answers
30 views

$A\alpha_1+\dots +A\alpha_n$ is a $B$-module

Let $A$ be an integrally closed domain with quotient field $K$ and $L/K$ a finite separable field extension. We denote with $B$ the integral closure of $A$ in $L$. Now let $\alpha_1,\dots,\alpha_n\in ...
-1
votes
1answer
29 views

Equivalence of a integer domain

If $R$ is a commutative ring with $1$. Suppose that for all polynomial $P(X)\in{R[X]\setminus{R}}$ has at most $n$ roots, with $n=grad\ (f)$ then $R$ is an integer domain. Any suggestion, please.
0
votes
1answer
35 views

Does the relation $\pi(S_{i})=S^{-1}R-P_{i}\cdot S^{-1}R$ hold for prime ideals $P_i$ in a commutative ring $R$?

Let $R$ be a commutative ring. Let $P_{i}$, $1\leq i\leq n$ be prime ideals none of which are contained in each other. Let $S=R-(\cup_{i=1}^{n} P_{i})$. Then $S$ is a multiplicatively closed set and ...
1
vote
2answers
42 views

Question on quotient ring

Let $\zeta := \zeta_p = e^{2\pi i/p}\in \mathbb{C}$ and set $R = \mathbb{Z}[\zeta] = \left\{\sum_{i=0}^{p-1} a_i\zeta^i\mid a_i \in \mathbb{Z}\right\}$. Let $\mathfrak{p} = (1-\zeta)$ be an ideal of ...
0
votes
2answers
61 views

In an Integral Domain, every prime is an irreducible. Flaw in the Proof?

In an Integral Domain, every prime is an irreducible. The proof is as follows : Let $D$ be the integral domain, then, if $a \in D$, it's possible to express a = $bc$ where $b,c \in D ...(1)$. ...
1
vote
0answers
42 views

Relative topological tensor product

I know that for a pair of topological rings $R$ and $R'$ (perhaps with some additional topological/functional-analytic hypothesis on them?) there exist two topological tensor produts, the injective ...
0
votes
3answers
44 views

Recovering Unique Factorization

Can a (commutative) ring $S$ fail to have unique factorization but be a subring of a (possibly noncommutative) ring $R$ which does have unique factorization? The idea being that the irreducibles in ...
1
vote
1answer
42 views

Ideal in a certain algebra over a field

Let $K|k$ be a finite field extension. Define $D$ to be a finite dimensional $k$ division algebra. If $J$ is a nonzero two-sided ideal of $D\otimes_k K$ then by considering $K$-dimensions, I see that ...
1
vote
1answer
86 views

How do we know that $f(x)\in Y$?

At page 19 in this book $f:X\to Y$ is defined to be $$f(a):=(\tilde\varphi(T_1')(a),\dots,\tilde\varphi(T_n')(a)).$$ To explain the notation above, $X\subseteq \mathbb{A}^m(k)$, $Y\subseteq ...
1
vote
2answers
135 views

Is $\mathbb{C}[x,y] / (y^2-x^3)$ a PID?

First, I'd like to show $\mathbb{C}[x,y] / (y^2-x^3)$ is an integral domain. Then I need to find out whether or not it is a PID. For the first part, I want to show $y^2-x^3 \: | \: fg \implies ...
6
votes
5answers
300 views

A finite commutative ring with 1 whose elements satisfy a particular equation

I would be very grateful if you give me a hint on it: Suppose $R$ is a finite commutative ring with identity such that $ x^3 = x $ for all elements $x$ of $R$. Then $R$ is a finite direct product ...
1
vote
2answers
36 views

An integral domain and its field of fractions.

I'm trying to solve the following problem: Let $R$ be a integral domain which is not a field and $K$ its fractions field. Show that a non-zero module $R$-homomorphism from $K$ to $R$ does not ...
3
votes
2answers
72 views

How to show this ring is not a UFD

I am trying to show that $R=\Bbb Z[x,y,z,w]/(xw-zy)$ is not a UFD. Let $I=(xw-zy)$. Let $X=x+I$, $Y=y+I$, $Z=z+I$, and $W=w+I$. My guess is that $X$ is irreducible and therefore $(X)$ is a ...
2
votes
2answers
82 views

Isn't $x^2+1 $ irreducible in $\mathbb Z$, then why is $\langle x^2+1 \rangle$ not a maximal ideal in $\mathbb Z[x]?$

Isn't $ x^2+1$ irreducible in $\mathbb Z$, then why is $\langle x^2+1 \rangle$ not a maximal ideal in $\mathbb Z[x]$? $ x^2+1$ cannot be broken down further non trivially in $\mathbb Z[x]$. hence, ...
2
votes
1answer
59 views

Can polynomials with degree at least 2 over $\mathbb{R}$ have finite number of solutions in $End(\mathbb{R},+)$

Consider a polynomial of degree at least 2 with all coefficients in $\mathbb{R}$. We are concern with set of solution for the polynomial in $End(\mathbb{R},+)$ - the endomorphism ring of the abelian ...
1
vote
1answer
42 views

Noetherian group rings

I'm asking for an example of a finitely generated amenable group $G$ and a field $K$, such that the group ring $K[G]$ is not Noetherian. Is it also possible to find a finitely generated amenable ...
1
vote
2answers
36 views

If $p$ is a prime prove that $x^{p-1} - x^{p-2} + x^{p-3}- \cdots -x+1$ is irreducible over $Q$

If $p$ is a prime prove that $x^{p-1} - x^{p-2} + x^{p-3}- \cdots -x+1$ is irreducible over $Q$. $1$st Attempt: $x^{p-1} - x^{p-2} + x^{p-3}- \cdots -x+1 $ $= x^{p-2}(x-1)+x^{p-4}(x-1)+ \cdots ...
0
votes
1answer
33 views

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 ...
3
votes
1answer
67 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 ...
2
votes
1answer
37 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 ...
-1
votes
1answer
48 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 ...
2
votes
3answers
46 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, ...
5
votes
3answers
186 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 ...
5
votes
3answers
83 views

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 ...
0
votes
1answer
33 views

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
votes
2answers
125 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 ...
1
vote
1answer
21 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 ...
0
votes
1answer
55 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 $f(x)$ ...
4
votes
0answers
43 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 ...
1
vote
1answer
21 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 ...
0
votes
2answers
28 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 $a^2=a$
3
votes
1answer
40 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 ...
0
votes
2answers
78 views

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 ...
2
votes
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 ...
0
votes
0answers
30 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, ...
1
vote
0answers
40 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 $R$ ...