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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21 views

Matrices representing a map between free modules of infinite rank and Fitting's Lemma (Eisenbud)

p.497 of Commutative Algebra with a View Toward Algebraic Geometry, Eisenbud: If $\phi: F \rightarrow G$ is a map of free modules, then $I_j\phi$ is the image of the map $$\Lambda^j F ...
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3answers
41 views

For which $n \in \mathbb{N}$ does $x^8 + [2]_nx^7+[6]_nx^6-x^2-[2]_nx - [6]_n = [0]_n$ have at least $7$ distinct solutions?

I have to find one $n \in \mathbb{N}$ such that $$x^8 + [2]_nx^7+[6]_nx^6-x^2-[2]_nx - [6]_n = [0]_n$$ has at least $7$ distinct solutions in $\mathbb{Z}_n$ (or, equivalently, $f(x) = x^8 + ...
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1answer
42 views

Correspondence between prime and maximal ideals [on hold]

My professor put the following statement in the lecture notes without proof: Let $R$ be a commutative ring and $I$ an ideal. Then the natural correspondence between ideals containing $I$ and ideals ...
1
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1answer
37 views

Intersection of two ideals

Let $A$ be a commutative ring and let $\mathfrak{a}$, $\mathfrak{b}$ be ideals in $A$. I am asked the following question: Show that $\mathfrak{a} \cap \mathfrak{b}$ is the largest ideal of $A$ ...
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2answers
19 views

Is the set of all $f$ such that $\lim_{x\to1^-}f(x) = 0$ an ideal in the ring of functions from $[0,1]\rightarrow \mathbb{R}$?

Is the set of all $f$ such that $\lim_{x\to1^-}f(x) = 0$ an ideal in the ring of functions from $[0,1]\rightarrow \mathbb{R}$? I'm sure about the closure under addition but not quite clear about if ...
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0answers
33 views

Generalisation of euclidean domains

Recently I wondered how dependent the definition of euclidean domains is of the co-domain of the norm-function. To be precise: Let's define a semi-euclidean domain as a domain $R$ together with a ...
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2answers
45 views

An irreducible polynomial in a subfield of $\mathbb{C}$ has no multiple roots in $\mathbb{C}$

Let $K\subset \mathbb{C}$ be a subfield and $f\in K[t]$ an irreducible polynomial. Show that $f$ has no multiple roots in $\mathbb{C}$. If I understand this question correctly, I must show that ...
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1answer
28 views

$f$ is divisible by a square of non-constant polynomial iff $f,f'$ are not relatively prime

Let $R$ be a commutative ring and $f=a_0+ \cdots +a_nt^n \in R[t]$. Define $f':=a_1+2a_2t+ \cdots + na_{n-1}t^{n-1}$. Show that $f$ is divisible by a square of non-constant polynomial if and only ...
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2answers
19 views

Union Over a Totally Ordered Set of Ideals is an Ideal

I am trying to understand the proof of a theorem which uses Zorn's lemma. I understand quite well all parts of the proof except for one point: Let $R$ be a ring and define $K\doteq \{I\subseteq R ...
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1answer
29 views

Does localization of a Noetherian ring always give a local ring? [on hold]

I have a local ring $A$ and suppose I localized this ring at prime $P$. Is the localized ring $A_P$ a local ring? I was wondering if it requires additional properties on $A$. Thank you very much!
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1answer
53 views

Property of a Noetherian ring: How come $P \setminus P^2$ is non-empty? ($P$ is a prime ideal) [on hold]

Let $A$ be a Noetherian ring, and let $P$ be a prime ideal. How come we know that $P \setminus P^2$ is non-empty? Thank you!
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1answer
28 views

Determining all the homomorphisms $\mathbb{Z} \to R$, where R is an integral domain.

I think I have this question figured out almost completely, but I'm a little worried about using a certain notation. Suppose $\mathbb{Z} \stackrel{\phi}{\longrightarrow} R$ is a ring homomorphism. ...
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1answer
12 views

Computing generators for a finitely generated module

I came across this problem yesterday: Let $R$ be a ring and $M$ an $R-$module. $\varphi:R^n\to M$ is a surjective $R-$module homomorphism if and only if $M$ is finitely generated. Given the set of ...
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1answer
70 views

When is an ideal also a ring, and could then be anything said about its relation to the original ring

If $R$ is a ring with unity $1$, then $S \subseteq R$ is called a subring if it is itself a ring with $1 \in S$. A subset $I \subseteq R$ is called an ideal if it is a group with respect to addition ...
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0answers
20 views

Show that we have a ring isomorphism $\varphi : D^{op} \rightarrow {End_{M_n {(D)}}}(D^n) $. [on hold]

I am trying to solve the following Representation Theory question: Suppose that $d \in D$ and define the map $$ \varphi_d \colon D^n \rightarrow D^n $$ by $$ \varphi_d((v_1, \ldots, v_n)) ...
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1answer
39 views

In proof of $|G| = n_1^2 + n_2^2 + \ldots + n_h^2$ how could equality of dimensions concluded from ring isomorphism

In Derek Robinson, A Course in the Theory of Groups on page 224 he proves: Let $G$ be a finite group and let $F$ be an algebraically closed field whose characteristic does not divide the order of ...
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0answers
43 views

Field of fractions of $\mathbb{C}[X,Y,Z]/\langle XY-Z^2\rangle$ [on hold]

Let $R=\mathbb{C}[X,Y,Z]/\langle XY-Z^2\rangle$ and let $\mathbb{C}(X,Y)$ be the field of fractions of $\mathbb{C}[X,Y]$. Show that the field of fractions of $R$ can be expressed as ...
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0answers
17 views

Prove that $\varepsilon(v) \equiv \varepsilon(u) \equiv 1 (2)$

Suppose I have a finite group $G$ and its integral group ring $\Bbb{Z}G$. Let $P < G$ , thus we have $\Bbb{Z}[C_G(P)] \subseteq \Bbb{Z}G$. Let $u\in U(\Bbb{Z}G)$ and let $v\in \Bbb{Z}[C_G(P)]$ be ...
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1answer
30 views

Embeddable rings axiomatic class?

In this question, a ring is defined to be with a unit distinct from the zero element, not necessarily a commutative ring though. Is the class of all such rings that can be embedded into fields an ...
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0answers
36 views

Reduction modulo $p$ of $x^4 + 3x^3 -21x^2 -62x -40$ with a multiple root

Let us consider $$g(x) = x^4 + 3x^3 -21x^2 -62x -40 \in \mathbb{Z}[x].$$ How does one find the primes $p>0$ such that the reduction of $g(x)$ modulo $p$ has a multiple root (without taking into ...
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0answers
28 views

Pick out a polynomial such that ideal $J=q(x)R$ , where $q(x)$ is polynomial and $R$ is ring [on hold]

In the ring of polynomials $R =\mathbb Z_5[x]$ with coefficients from the field $\mathbb Z_5$, consider the smallest ideal $J$ containing the polynomials, $p_1(x) = x^3 + 4x^2 + 4x + 1$ $p_2(x) = ...
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1answer
58 views

Is I-adic completion a ring epimorphism?

Let $R$ be a commutative ring and let $I \subset R$ be an ideal. For any $n \ge 1$, the ring homomorphism $R \rightarrow R/I^n$ is surjective, hence an epimorphism in the category of rings. What about ...
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2answers
63 views

Affine varieties and their ideals

I was reading on Wikipedia about quotient ideals. It mentions that if $W$ and $V$ are affine varieties (assume $V$ is) and $I(V)$ and $I(W)$ are the ideals for $V$ and $W$, then $$I(V):I(W) = ...
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1answer
24 views

How do you prove a valuation ring is a subring?

Let's say I have a field $\mathbb{F}$. Now suppose I take the set $R = \{x \in \mathbb F^{\times}: \ y(x) \ge 0\} \cup \{0\}$ where $y$ is a function $y:\mathbb F^{\times} \rightarrow \mathbb{Z}$ ...
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1answer
48 views

How many elements are there in $\mathbb{Z}[\sqrt{2}]/(1+3\sqrt{2})$?

Let $\mathbb{Z}[\sqrt{2}]:= \lbrace a+b\sqrt{2}|a,b \in \mathbb{Z} \rbrace$. How many elements are there in $\mathbb{Z}[\sqrt{2}]/(1+3\sqrt{2})$? I know that every equivalence class of ...
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2answers
29 views

Proving that a homomorphism between two rings is surjective

The problem: ($\mathscr{F}(\mathbb{R})$ is the set of real valued functions) Let $\phi:\mathscr{F}(\mathbb{R})\to\mathbb{R}\times\mathbb{R}$ be a function defined by $\phi(f)=(f(0),f(1))$ Prove ...
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0answers
42 views

Show that $(2,1+\sqrt{-5})$ is a maximal ideal in $\mathbb{Z}[\sqrt{-5}]$ [duplicate]

Let $R=\mathbb{Z}[\sqrt{-5}]$ and $I=(2,1+\sqrt{-5})$ an ideal generated by $2$ and $1+\sqrt{-5}$. Show that $I$ is a maximal ideal. So I tried to prove that if $a \notin I$ then $(I,a)$ must be ...
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0answers
26 views

Groebner basis and prime ideals.

Let $I$ be an ideal in a polynomial ring $K[x,y_1,\dots,y_n]$ and assume that $I \cap K[x]\neq (0)$. Let $>$ be an elimination ordering for $\{y_1, \dots, y_n\}$ and $G$ is a Groebner basis for $I$ ...
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1answer
58 views

Finite number of maximal ideals of bounded norm

Suppose that we have an integral extension of rings $R\subseteq S$ and $S$ is finitely generated as $R$-module or as $R$-algebra, and $R/\mathfrak m$ is finite for all maximal ideals and $S/\mathfrak ...
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1answer
49 views

There exists a zero dimensional ideal I such that $\dim (R/I) - |V(I)| \geq \alpha > 0$

If $I \subset K[x_1,\dots,x_n]$ is a zero dimensional ideal and $$V(I) = \{ (\alpha_1,\dots,\alpha_n) \in K^n: f((\alpha_1,\dots,\alpha_n)) = 0\ \forall f\in I\}$$ (the variety). I know that ...
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0answers
15 views

A sufficient condition for factorization in a complete local ring

I think something like the following statement is true, but I don't recall a reference. Suppose $f(x,y)\in k[[x,y]]$ is power series with no constant or linear terms. Then, if the quadratic terms ...
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0answers
16 views

Proof that maximal ideals in $\mathcal{P}[x_0]$ intersected with $\mathcal{P}$ is a maximal ideal in $\mathcal{P}$ [on hold]

I am trying to show that maximal ideals in $\mathcal{P}[x_0]$ intersected with $\mathcal{P}$ is a maximal ideal in $\mathcal{P}$, where $\mathcal{P}$ is the polynomial ring $K[x_1, \dots, x_n]$ or ...
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1answer
35 views

Quotient of maximal and prime ideals [on hold]

Given that $I, J$ are ideals in $R$, $I$ is maximal or prime, do we have that $I/J$ is maximal in $R/J$? $I/J$ is prime in $R/J$? I think it is true but don't see how it works.
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2answers
48 views

Two ways to decompose $\mathbb Q[G]$ for $G = \{1,g,g^2\}$ and their interrelation.

Let $G = \{1,g,g^2\}$ be a cyclic group of order $3$. Consider the group ring $\mathbb Q[G]$. Then we have the two $G$-invariant simple subspaces $$ I = \{ a_0 + a_1 g + a_2 g^2 : a_0 = a_1 = a_2 \} ...
3
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1answer
51 views

Suppose A is a principal ideal domain with every ideal of finite index. Must A be a Euclidean domain?

Suppose $A$ is a principal ideal domain with every ideal of finite index (except the zero ideal). Must $A$ be a Euclidean domain? If it's not known, are there any relevant partial results?
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2answers
52 views

If $G \cong \mathbb Z/3\mathbb Z$, then $\mathbb R[G] \cong \mathbb R \times \mathbb C$

Let $G = \{1,g,g^2\}$ be the cyclic group of order three. Consider the group ring $\mathbb R[G]$, then $\mathbb R[G] \cong \mathbb R \times \mathbb C$ with the isomorphism $$ \varphi(1) = (1,0), ...
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1answer
53 views

Non-Euclidean domains examples which have universal side divisor

Let $R$ be a ring. A nonzero nonunit element of $R$ is called a universal side divisor if for every element $x$ of $R$ there is some element $z$ of $R$ such that $u$ divides $x - z$ in $R$ where $z$ ...
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1answer
62 views

Explanation of a proof about graded module structure

Let $\Bbb F$ be a field and $M$ a finitely generated $\Bbb F[x]$-module. The structure theorem for modules over a PID says that $$ M\cong \Bbb F[x]^r\oplus\biggl(\bigoplus_{j=0}^s\Bbb ...
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2answers
53 views

Exercise concerning quotient rings and ideals

I need help proving the following: Let $A$ and $B$ be rings with $\beta\in B\subseteq A$ and suppose the ideal $\beta A$ is a prime ideal. If the "/" denotes the quotient set and the equality ...
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1answer
40 views

Is ${R /I} \cong {{\mathbb Z} /{9\mathbb Z}}$? [on hold]

If $$R = \mathbb Z[\sqrt2] = \{a+b\sqrt2\mid a,b \in \mathbb Z\}\\I = \{a+b\sqrt2\mid a,b \in 3\mathbb Z\}$$ Is ${R /I} \cong {{\mathbb Z} /{9\mathbb Z}}$?
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2answers
60 views

Show that for a commutative ring $R$, there always exists a ring homomorphism from $R$ to some field.

Show that for a commutative ring $R$, there always exists a ring homomorphism from $R$ to some field. I try to extend $R$ to a division ring by throwing in multiplicative inverse for each ...
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1answer
35 views

Example of Artinian module which has infinitely many maximal submodules not isomorphic to each other

I'm looking for an Artinian module which has infinitely many maximal submodules not isomorphic to each other. I guess I can find it over a matrix ring.
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1answer
87 views

Examples of non-principal free ideals

If $R$ is a commutative ring, and $I\subset R$ is a non-zero free ideal, then it is principal generated by a non-zerodivisor. If $R$ is a non-commutative ring having the IBN property and $I\subset R$ ...
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1answer
108 views

Example of a ring such that $R^2\simeq R^3$, but $R\not\simeq R^2$ (as $R$-modules)

The usual example of unitary ring without the IBN property is the ring of column finite matrices, and in this case we have $R\simeq R^2$ as (left) $R$-modules. (See also here.) In particular, we have ...
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1answer
48 views

Exercise from Kaplansky - Commutative Rings (1.2.3)

Exercise 3 in section 1-2: Let $R$ be an integral domain, $P$ a finitely generated non-zero prime ideal in $R$, and $I$ an ideal in $R$ properly containing $P$. Let $x$ be an element in the ...
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3answers
84 views

How many zero divisors are in a finite ring?

I'd like to know how many zero divisors there are in the ring $\mathbb{Z}_5[x]/(x^3-2).$ Is it sufficient to note that for $p(x) = x^3-2,$ $p(3) = 0$ and $p(5)=0?$ I haven't been able to write ...
1
vote
1answer
29 views

Relating generating sets of modules over quotient ring to modules over original ring

I am having trouble with the following exercise. Let $R = F[x]$, where F is some field. Let $I = (x^2)$, and let M be an R/I-module (induced action, so assuming $IM = 0$). Then I want to show that ...
2
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2answers
40 views

Does an algebra over a ring with unity have unity?

Definition from ProofWiki: An algebra over a ring $(G_R,\oplus)$ is an $R$-module $G_R$ over a commutative ring $R$ with a bilinear mapping $\oplus:G^2→G$. Context: I want to show that an algebra ...
0
votes
1answer
31 views

Symbolic power of a prime ideal is primary

Let $A$ be a commutative ring, $S$ a multiplicatively closed subset of $A$. For any ideal $\mathfrak a$, let $S(\mathfrak a)$ denote the inverse image of $S^{−1}\mathfrak a$ under the localization map ...
3
votes
2answers
54 views

$\overline{Y}$ is irreducible in $A:=\Bbb{C}[X,Y]/ (X^2-Y^3)$. [duplicate]

I would like to prove that $A:=\Bbb{C}[X,Y]/ (X^2-Y^3)$ is not a UFD. This is equivalent to find an irreducible element which is not prime. I can prove that every element looks like ...