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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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
23 views

Does localization of a Noetherian ring always give a local ring?

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
44 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
18 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
10 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
34 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
18 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
38 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
35 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
13 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
28 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
24 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
53 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
59 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
19 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
42 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
28 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
38 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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19 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
40 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
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
39 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
27 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
22 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
15 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
13 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 $p$ ...
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1answer
32 views

Quotient of maximal and prime ideals

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
68 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
47 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 \} ...
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1answer
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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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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 ...
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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.
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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
49 views

Why is this a faithfully flat extension?

How can I see that the map of rings $f \colon A = \mathbb{Z}_{(3)}[a_2,a_4,a_6] \to B = \mathbb{Z}_{(3)}[a_2,a_4,r]$ given by $a_2 \mapsto a_2 + 3r$, $a_4 \mapsto a_4 + 2a_2 r+ 3r^2$ and $a_6 ...
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2answers
40 views

Maximal and Prime Ideals

I was assigned these problems for homework to designate if they were maximal, prime or neither. I was able to determine that (a) was solely prime by showing $\mathbb{Z}[x] /(x-1)$ is isomorphic to ...
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2answers
37 views

About radical of $(I,x)$ with $x$ irreducible

Let $I$ be a proper ideal of a polynomial ring $A$ and $x \in A$ an irreducible element. In a theorem of commutative algebra I will use the fact that, in this hypothesis, holds the following ...
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1answer
464 views

When Leavitt path algebras are unital

I’ve been taking a seminar course on Leavitt path algebras, and our source material is rather laconic as far as explanations are concerned. I am attempting to justify (to myself) the following claim ...
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1answer
896 views

Correspondence theorem for rings.

Could someone provide a reference that includes a full and honest proof of the Correspondence Theorem for rings? Let $A$ be a multiplicative ring with identity and $I$ an ideal of $A$. There is a ...
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0answers
44 views

Brauer Group - A measure of complexity?

I have seen many authors state that the Brauer Group in some way measures the complexity of a field. I've convinced myself that the Brauer group of the reals is Z/2Z, and that the Brauer group of an ...
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2answers
413 views

Can the product of two non invertible elements in a ring be invertible?

Let $A$ be a unitary ring. The question is simply: can the product of two non invertible elements in $A$ be invertible? I proved that the answer is negative if $A$ does not have zero divisors, ...
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vote
1answer
62 views

When is a simple tensor equivalent to natural multiplication?

Let $R$ be a commutative integral domain; let $M, N$ be $R$-modules. Let $x_1 \in M$ and let $x_2 \in N$. When does the following equivalence hold? $$x_1 \otimes x_2 = x_1x_2$$ The textbook I'm ...
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1answer
1k views

Show that every ideal of the matrix ring $M_n(R)$ is of the form $M_n(I)$ where $I$ is an ideal of $R$ [duplicate]

Suppose $R$ is a commutative ring. Show that every ideal of $M_n(R)$ is of the form $M_n(I)$ where $I$ is an ideal of $R$. I have spent 30 minutes on this question and I still got nowhere. Can ...
4
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3answers
356 views

can the square of a proper ideal be equal to the ideal

Let $R$ be a ring, commutative with $1$, let $\mathfrak{i}$ be an ideal, not the whole ring. In general $\mathfrak{i}^2\subseteq\mathfrak{i}$. Can this inclusion be an equality, or it is always a ...
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3answers
2k views

What are applications of rings & groups?

I am following a course in basic algebra, and we have covered rings & groups in class, but I am having trouble visualising them. Are there applications of group &/or ring theory that can be ...