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
27 views

Noncommutative rings and the evaluation homomorphism

Recall the evaluation homomorphism of a ring. For example, if $\{R[x]|{p(x)=a_0+a_1x+a_2x^2...}$} is the ring of polynomials with real coefficients then we can evaluate with respect to $c$ by letting ...
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0answers
22 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
42 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 + ...
0
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1answer
43 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!
1
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1answer
29 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. ...
0
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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 ...
1
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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)) ...
3
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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 $y:\mathbb F^{\times} \rightarrow \mathbb{Z}$ ...
2
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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 ...
2
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1answer
25 views

If $a \ne b$ in a ring $R$ satisfy $a^3 = b^3$ and $a^2b = b^2a$, show that $a^2 + b^2$ is not a unit.

If $a \ne b$ in a ring $R$ satisfy $a^3 = b^3$ and $a^2b = b^2a$, show that $a^2 + b^2$ is not a unit. So I am thinking that I should be able to do this by contradiction. So if I assume there is some ...
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1answer
50 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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2answers
18 views

If char$R=n$, show that $\mathbb{Z}1_R\cong \mathbb{Z}_n$.

Let $1_R$ be the identity of a ring $R$ and let $\mathbb{Z}1_R=\{k1_R\mid k\in\mathbb{Z}\}$. If char$R=n$, show that $\mathbb{Z}1_R\cong \mathbb{Z}_n$. So my thought is I just have to think of some ...
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1answer
28 views

Why is $|V(I)| \leq d_1\cdots d_n$?

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). Then if $G$ is a ...
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3answers
27 views

If $S$ and $T$ are subrings of $R$, is $S+T$ a subring of $R$?

If $S$ and $T$ are subrings of $R$, is $S+T=\{s+t\mid s\in S, t\in T\}$ a subring of $R$? So I think that $S+T$ is a subring, but I am getting stuck trying to prove it. Clearly since $S$ and ...
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2answers
20 views

Clarify what “inclusion preserving” means in lattice isomorphism theorem

I'm working through Dummit and Foote right now. The lattice isomorphism theorem is stated as follows: "Let I be an ideal of a ring R. The correspondence $A \leftrightarrow A/I $ is an inclusion ...
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0answers
23 views

why is the collection of all finite subsets of $\mathbb{R}$ not a $\sigma-ring$

It says the definition of a $\sigma-ring$ is if $A,B \in \mathcal R$ then $A \setminus B \in \mathcal R$ and if $ A_{n} \in \mathcal R \forall n \in \mathbb{N}$ then $\cup_{1}^{\infty}A_{n} \in ...
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1answer
16 views

Relationship between operations of a ring

Is there any requirement that the two operations of a ring have to be related to each other, excluding the requirement of distributivity? We all know from grade school that multiplication of integers ...
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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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1answer
10 views

factors in polynomial rings with field coefficients

I'm reading through Dummit and Foote Abstract Algebra, and I was looking for an explanation of the following: Proposition 9: $F$ a field and $p(x)\in F[x]$. Then $p(x)$ has a factor of degree one if ...
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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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3answers
27 views

A nontrivial solution to the quadratic form $x^2 - xy + y^2$ over the finite field $𝔽_p$ with $p ≡ 1 \pmod3$ a prime

I'm trying to prove that when $p ≡ 1 \pmod3$ is a prime, $p$ is reducible over the Eisenstein integers, and I've gotten to the point where, provided $p\,|\,u^2 - u + 1$ for some integer $u$, then $p$ ...
0
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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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3answers
70 views

Algebraically, why is $\mathbb{Z}[i]/(i + 1)$ isomorphic to $\mathbb{Z}_{2}$? [duplicate]

I understand geometrically why the Gaussian integers modulo $i+1$ is $\mathbb{Z}_{2}$, using lattices. Is there an algebraic isomorphism construction, however?
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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
votes
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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0answers
21 views

The coordinate ring of $\varepsilon: xy-1=0$ [duplicate]

I want to show that the coordinate ring $\mathbb{R}[x,y]/\mathbb{R}[\varepsilon]$ of $\varepsilon: xy-1=0$ is not isomorphic with the polynomial ring of one variable $\mathbb{R}[x]$. To me this is ...
0
votes
1answer
11 views

Addition of fractions in z11

compute 3/5+2/7+1/6 in Z11. Please give me a hint on how to go about it. I have created a table for Z11 but unsure of the next step.
0
votes
1answer
81 views

Algebraic element - integral domain

Let $K$ a field and $L$ a subfield of $K$. Let the set $\overline{L}:= \{k \in K: k$ is algebraic over $L$ $\}$ is another subfield of $K$. Show that $\overline{\overline{L}}=\overline{L}$. ...
3
votes
2answers
50 views

Prove or disprove : the number $3+2\sqrt{-2}$ is irreducible in the ring $\mathbb{Z}[\sqrt{-2}]$

Prove or disprove: the number $3+2\sqrt{-2}$ is irreducible in the ring $\mathbb{Z}[\sqrt{-2}]$. I think it is sufficient to show that each element (except $0$) in $\mathbb{Z}\sqrt{D}$ with $D ...