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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1answer
54 views

Consider the ring $R=\mathbb{Q}[x]$

$\newcommand{\QQ}{\mathbb{Q}}$ Consider the ring $R=\QQ[x]$ over $I$ where $I$ is an ideal of $\QQ$ generated by the polynomial $x^2-2$ in $\QQ[x]$. (a) Show that for all $f(x)$ in $\QQ[x]$, ...
1
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1answer
53 views

I as an ideal of $R$ then $a+I=0+I$ iff $a\in I$ [duplicate]

show that if a,b belong to the ring $R$ and $I$ is an ideal of $R$ then $a+I=0+I$ if and only if $a$ belongs to $I$. I know that since I is an ideal then it is both a left and a right ideal.
2
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4answers
92 views

In the proof of “maximal ideals of $\Bbb{Z}[x]$ are of the form $(p,f(x))$ ”

I'm trying to prove the following statement: Maximal ideals of $R=\Bbb{Z}[x]$ are of the form $(p,f(x))$ where $f(x)$ is irreducible in $\Bbb{F}_p[x]$ and $p$ is prime. A quick search on google ...
3
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3answers
421 views

Nontrivial ideal of a Noetherian domain contains a finite product of nonzero prime ideals

If $R$ is a Noetherian domain and $ 0 < U < R$ an nontrivial ideal of $R$. How to prove that there exists nonzero prime ideals $p_1,...,p_n \subset R$ such that the product $ p_1 p_2 ...p_n ...
1
vote
1answer
45 views

Epimorphism that is not monomorphism from $M\rightarrow M$

I have just finished an exercise where I prove that if $M$ is a module with acc then any epimorphism $f:M\rightarrow M$ but be an isomorphism. I then had a think about examples of non-noetherian ...
2
votes
2answers
165 views

Maximal ideal of $\Bbb Z$ that is not maximal in $\Bbb Z[X]$

Can someone come up with an example of a maximal ideal P in $\mathbb{Z}$ such that P[X] is not maximal in $\mathbb{Z}[X]$ - the ring of polynomials with integer coefficients? I know that the maximal ...
2
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0answers
227 views

LCM generators for the intersection of non-principal ideals in a Noetherian UFD

I am working with some non-principal ideals $I=\langle a,b\rangle$, $J=\langle c,d\rangle$ in a nicely behaved Noetherian UFD (the Laurent polynomial ring in finitely many commuting variables with ...
7
votes
1answer
334 views

Automorphisms of the ring $\Bbb Z[x]$ of polynomials with integer coefficients

I am trying to find the automorphisms of the polynomial ring $\mathbb{Z}[x]$. So far, I have shown that if $\varphi$ is an automorphism, then $\varphi(n)=n$ for $n \in \mathbb{Z}$, and $\varphi(x)$ ...
1
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1answer
1k views

Prove that the Gaussian rationals is the field of fractions of the Gaussian integers

I'm looking to prove that $\Bbb Q[i] = \{ p + qi : p, q \in \Bbb Q \}$ is the field of fractions of $\Bbb Z[i] = \{p + qi : p, q \in Z\}$. I am familiar with definition of a field of fractions. For ...
0
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1answer
103 views

What is needed to force polynomial-ring automorphisms to be affine?

Is there an integral domain $R$ and a polynomial-ring automorphism $\: \phi : R[x] \to R[x] \:$ such that, for $\: i : R\to R[x] \:$ the canonical embedding, $\;\;\; \phi \circ i \: = \: i \;\;$ and ...
3
votes
2answers
267 views

What is the proof of the single factor theorem over an arbitrary commutative ring?

Theorem (Single factor theorem) Let $R$ be a commutative ring, and let $P\in R[X]$, where $R[X]$ is the polynomial ring over the indeterminate $X$. Suppose $P(\alpha)=0$. Then $(X-\alpha)$ divides ...
1
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1answer
65 views

Transcendental number over $\{k\in K\mid f(k)=k\}$

Let $K$ be a field and $f:K\rightarrow K$ be a ring endomorphism. Prove that if $\alpha\in K\setminus f(K)$, then $\alpha$ is transcendental over the subfield of $K$, $F:=\{k\in K\mid f(k)=k\}$. My ...
2
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1answer
110 views

Can this quick way of showing that $K[X,Y]/(Y-X^2)\cong K[X]$ be turned into a valid argument?

I've been trying to show that $$ K[X,Y]/(Y-X^2)\cong K[X] $$ where $K$ is a field, $K[X]$ and $K[X,Y]$ are the obvious polynomial rings over the indeterminates $X$ and $Y$ and $(Y-X^2)$ is the ...
1
vote
1answer
109 views

Trying to prove pre image of product of ideals is, the product of the pre images of the two ideals…

This is from Elements of Abstract Algebra by Allan Clark. 166 $\beta$ $\Phi ^{-1}(a'b') = (\phi^{-1}(a'))(\phi^{-1}(b'))$ I can prove an element of the right side is an element of the left. But I ...
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1answer
284 views

Intersection of Two Ideals in the Ring of Integers

I'm working through this question If the intersection of $x\mathbb{Z}$ and $y\mathbb{Z}$ equals $w\mathbb{Z}$ for positive integers $x$, $y$, and $w$ (ideals in ring $\mathbb{Z}$), express the ...
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2answers
77 views

Unital rings within matrices

Let $R$ be a commutative, unital ring. Define $$ R[\mathbf{t}]= \left\{ \begin{bmatrix} \mathbf{w} & \mathbf{z} \\ -\mathbf{z} & \mathbf{w}-\mathbf{z} \end{bmatrix}\in R_2^2\;\middle|\; ...
7
votes
1answer
249 views

Idempotents in a ring without unity (rng) and no zero divisors.

Question: Given a ring without unity and with no zero-divisors, is it possible that there are idempotents other than zero? Def: $a$ is idempotent if $a^2 = a$. Originally the problem was to ...
0
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0answers
60 views

Show $\mathbb F_{p}[x]/(x^{p}-1)$ is indecomposable as a representation of $\mathbb Z/ p\mathbb Z$

Let $R=\mathbb F_{p}[x]/(x^{p}-1)$. $R$ has both ring and vector space structure. I am trying to show that, given a representation $\rho : \mathbb Z/ p\mathbb Z\rightarrow GL(R)$, any invariant ...
6
votes
2answers
309 views

The maximal ideal in a local ring is finitely generated

Assume $m<R$ is the maximal ideal of a commutative local ring with identity, such that $m=m^2$. Is $m$ finitely generated? Is the condition $m=m^2$ redundant? I am trying to apply Nakayama's lemma ...
3
votes
1answer
59 views

Division ring as a $K$-algebra.

I want to solve the following question: Suppose that the division ring $\Delta$ is a $K$-algebra with $(\Delta:K)$ finite. Prove that $\Delta=K$ if $K$ is algebraically closed. Deduce that if $K$ ...
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3answers
464 views

Factoring a polynomial in a field into irreducible

Factor $x^3 + 2x + 3$ into irreducible polynomials in $\mathbb{Z} _5 [x]$ This polynomial has 2 zeros mod 5: x = 2 and x = 4. But these only give me a 2 degree polynomial $x^2 - 4$ and I don't know ...
1
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1answer
94 views

Endomorphism ring, module, radical

For a module $C$ and over a ring $R$, $E:=\operatorname{End}_R(C)$, if $1-x$ is in $\operatorname{Rad}(E)$,the intersection of all left maximal ideals, then there is $v$ in $E$ such that ...
3
votes
2answers
49 views

Is it possible that a scalar times a submodule is not a submodule?

Does there exist a ring $A$ and an $A$-module $X,$ such that for some $a \in A$ and some submodule $Y \subseteq X$, it holds that $aY$ is not a submodule? If $A$ is commutative, this is clearly ...
3
votes
1answer
150 views

Polynomial is zero for induced mapping of rings

Let $R$ be a commutative ring, and $M$ a finitely-generated free $R$-module. Let $\phi:M\rightarrow M$ be an $R$-linear map, and $P_\phi(X)$ the characteristic polynomial of $\phi$. Let ...
2
votes
1answer
76 views

Matrix algebras

Let $k$ be any field, then we know that every finite dimensional semi simple algebra $A$ is isomorphic to a direct product of matrix algebras with entries over a division ring. Assume that we require ...
3
votes
1answer
110 views

Counterexample for $A[[x, y]] = A[[x]][[y]]$

Maybe this is an idiot question, but I've heard that $A[[x, y]] = A[[x]][[y]]$ does not hold for $A$ an arbitrary commutative ring with identity, so I would like to know a counterexample, since the ...
0
votes
1answer
43 views

Given a polynomial $p(x)$ in $\mathbb Z_6[x]$, it is possible to construct a ring $R$ such that $p(x)$ has a root in $R$.

Prove or disprove: Given a polynomial $p(x)$ in $\mathbb Z_6[x]$, it is possible to construct a ring $R$ such that $p(x)$ has a root in $R$. For this exercise I think about complex numbers. ...
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0answers
164 views

Rings, annihilators and (maximal) ideals

Let $R$ be a unital, associative, non-commutative ring. If $P$ is an ideal of $R$, what is the annihilator of quotients $R/PR$ and of $R/P$? Does something change if $P$ is supposed to be a maximal ...
2
votes
1answer
270 views

Union of ascending ideals is an ideal

Could you tell me what I'm doing wrong in proving this proposition? If $I_1 \subset I_2 \subset ... \subset I_n \subset ...$ is an ascending chain of ideals in $R$, then $I := \bigcup _{n \in ...
6
votes
1answer
95 views

Integral ring extension

Let $R=k[t]/(t^2)$ and $S=k[t,x]/(t^2,tx^3+tx^2-x^2-x)$, $k$ is a field. I must prove that $S$ is integral over $R$ and that $S=R\oplus R$. Any help about that..thanks in advance...
2
votes
1answer
175 views

Triangular Matrices and Simple Modules

Let $\Bbb{T}_n(k)=\{n \times n \text{ upper triangular matrices (which includes the diagonal entries)}\}$ I want to first express $\Bbb{T}_{n}(k)$ (as a $\Bbb{T}(k)$-module) as a direct sum ...
3
votes
1answer
78 views

Radical ideal of leading terms and Grobner

Let $k$ be a field, let $A$ be an ideal of $k[x_1,\ldots,x_n]$, and let $>$ be a monomial order. I'm asked to show that $A$ is radical if $\langle LT(A)\rangle$ is radical. So, suppose $\langle ...
1
vote
1answer
93 views

Kernel and direct sum

Let $R=k[x_1,\ldots,x_7]$ be a polynomial ring over field $k$ and $I=\bigcap_{i=1}^4 \mathfrak{p}_i$ where $\mathfrak{p}_1=(x_1,x_3,x_5,x_6), \mathfrak{p}_2=(x_1,x_3,x_4,x_6), ...
2
votes
2answers
91 views

Computing kernel

Let $I,J$ be two ideals of Noetherian ring $R$. How to compute kernel of following homomorphism directly: $$\phi: R/I\oplus R/J\to R/(I+J) $$ $$(a+I,b+J)\to (a-b)+I+J $$
8
votes
1answer
308 views

Is $\mathbb{Z}[x] / \langle (x^2 + 1)^2 \rangle$ isomorphic to a familiar ring?

The quotient ring $\mathbb{Z}[x] / \langle (x^2 + 1)^2 \rangle$ was brought up in class today to contrast it with $\mathbb{Z}[x] / \langle x^2 + 1 \rangle$ after a discussion about adjoining elements ...
4
votes
1answer
84 views

Short exact sequences and finite injective dimension

Say that $0 \to M \to N \to L \to 0$ is a short exact sequence of modules in a Noetherian local ring and that inj dim$(M)$, inj dim$(N) < \infty$. Does this imply that $L$ also have finite ...
0
votes
1answer
77 views

Subring of Gaussian integers has no greatest common divisor property [duplicate]

Problem is: Produce elements a and b in the domain $R := \{x+2y\sqrt{-1} \mid x, y \in \mathbb{Z}\}$ having no gcd. How can produce this? Actually I use norm function, and brute force, but what ...
4
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0answers
124 views

How to prove the ring $R_J$ is Noetherian without use the theorem of I. S. Cohen?

First define $R:=k\left[{\{X_i\}}_{i\in\mathbb{N}}\right]$ where $k$ is a field (it could be an integral domain as $\mathbb{Z}$ too for example). This ring is an integral domain and it is not ...
3
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2answers
60 views

Maximal Ideals in the Integers

We know that the maximal ideals in $\mathbb{Z}$ are all ideals of the form $(p)$, where $p$ is prime. But what if we consider $(p, p_2)$, where $p_2$ is prime, although I'm not sure that the ...
2
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2answers
144 views

Noetherian Ring where all primes are maximal

I am trying to show that a commutative noetherian ring where all prime ideals are maximal is artinian. I know that every ideal contains a finite product of primes and also as all primes are maximal ...
0
votes
1answer
72 views

How to prove that $J(M_n(R))=M_n(J(R))$?

How to prove that $J(M_n(R))=M_n(J(R))$? Here $M_n(R)$ is the ring of matrices of size $n^2$ over the ring $R$. And $J(M_n(R))$ is a two-sided ideal of the ring $M_n(R)$.
3
votes
1answer
134 views

Find two elements that don't have a gcd in a subring of Gaussian integers

Find two elements in the domain $R := \{ x + 2y \sqrt {-1} \mid x,y \in \mathbb{Z} \}$ that do not have a gcd. I have no idea how to start. But I know if we consider $R^\prime = \{ x + y \sqrt ...
5
votes
2answers
1k views

Interview preparation for Ph.D admission

I have recently gave a written exam for admission to Ph.D program to an institute in India. I have done that exam well and hoping for an interview call. I would like to know what could be type of ...
10
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1answer
150 views

Is it really unknown that every endomorphism of the Weyl algebra $A_1$ is an isomorphism?

Here $A_1 := K\{x\cdot-, \frac{d}{dx}\} \subset \operatorname{End}_K(K[x])$ for some characteristic-zero field $K$. I found this claim in Coutinho's "A Primer of Algebraic D-Modules." If this is ...
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vote
2answers
200 views

Showing 3 is an irreducible element of $\mathbb Z[\sqrt{2}]$

What I tried: $$(3)=\{3r|r\in \mathbb Z\}\space\mbox{is a maximal ideal of}\space\mathbb Z\implies(3)=\{3(a+b\sqrt{2})|a,b\in\mathbb Z\}\space\mbox{is a maximal ideal of }\mathbb ...
1
vote
1answer
127 views

Localization by a non-nilpotent as a polynomial quotient

In Dummit and Foote, Exercise 15.18 (p. 727 3rd edition) one is asked to prove that for a commutative ring with unity $R$, and $f$ a non-nilpotent element of $R$, that the localization $R_f$ is ...
3
votes
2answers
111 views

Some isomorphisms of quotient rings.

Under which conditions on the ring $A$ do we have the isomorphism $A[x]_x\cong A[x,y]/(xy-1)$, and why does this even hold? I am asking because of the following isomorphism: ...
3
votes
2answers
153 views

The ring of integers of $\mathbf{Q}[i]$

Is there a relatively "simple" (in the sense that it does not require knowledge of algebraic number theory) proof that the ring of integers of the algebraic number field $\mathbf{Q}[i]$ is ...
7
votes
2answers
138 views

Ring of entire functions is integrally closed or not?

Is the ring $\mathscr{O}(\mathbf{C})$ of entire functions integrally closed (in its field of fractions, the meromorphic functions)? I know it's not factorial, but this doesn't exclude the ...
2
votes
1answer
63 views

Localizations of the ring $A=\Bbb{C}[X,Y]/(Y^2-X^4+2X^2-1)$

Consider the ring $A=\Bbb{C}[X,Y]/(Y^2-X^4+2X^2-1)$. Let $A_M$ denote the localization of $A$ with respect to maximal ideal $M$. My question is: Is $A_N$ a DVR, where $N$ is the maximal ideal ...