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
134 views

Ring structure on K-theory of an even-dimensioned sphere

I am slightly confused by some statements in Hatcher's Vector Bundle book (page 60). To start with (and I am happy with) the natural ring homomorphism $$K(S^2) \simeq \mathbb{Z}[H]/(H-1)^2$$ I am ...
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3answers
1k views

How to solve polynomial equations in a field and/or in a ring?

I'm studying for my exam, and I stuck on solving polynomials in a field and/or in a ring. Let me give you some examples: (1) Solve equation $x^2+4x+3=0$ in field $\mathbb{Z}_5$, $\mathbb{Z}_8$ and in ...
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1answer
337 views

Rings and modules of finite order

Fields of finite order are well classified, and classification of groups of finite order has taken some depth in research. Why classification of finite rings and modules is not well studied in ...
4
votes
1answer
135 views

Ring structure of K-theory of a wedge of spheres

I've just been using Bott Periodicity to calculate the K-theory of some simple spaces - spheres, torus, and wedge of spheres. The wedge of spheres is interesting. Given that $$\tilde{K}(X \vee Y) = ...
1
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1answer
127 views

writing a field as an R module

let $F$ be a field. for which ring $R$, $F$ is an $R$-module. i know already that as an abelian group $F$ is a $\mathbb Z$- module, what else can we say for a general field $F$.
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4answers
411 views

Can we make an integral domain with any number of members?

Is it true that rings without zero divisors (integral domains) can have any number of members except for 4,6? and if this is true then what would the multiplication operator be?
3
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0answers
88 views

Azumaya algebra and its subalgebras

I remind you that an Azumaya algebra $A$ is a central and separable algebra. Now, I know that if $A$ is an algebra over a skew-field or over a local ring then there exists a subalgebra $S$ of $A$ such ...
3
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2answers
165 views

Algebraic integers

Could someone please explain to me why $ \mathbb{F}_p [X] / \langle\bar{f_\alpha} (X)\rangle \,\ \cong \,\ \mathbb{Z}[X] / \langle p, f_\alpha (X) \rangle \,\ \cong \,\ \mathbb{Z} [\alpha] / \langle ...
4
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1answer
187 views

Can I assume $R$ is a local ring?

I should prove this statement: Let $R$ be a ring, $M$ be a $R$-module and $P$ a projective $R$-module of finite type. If $x=\sum_i m_i\otimes p_i$ is an element in $M\otimes_R P$ such that $\sum_i ...
1
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2answers
79 views

Is $\mathbb{Q}[[v^{-1}]] \cap \mathbb{Q}(v)=\mathbb{Q}[[v^{-1}]]$?

I have a question about the some rings and fields. Is $\mathbb{Q}[[v^{-1}]] \cap \mathbb{Q}(v)=\mathbb{Q}[[v^{-1}]]$?
4
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4answers
3k views

Can you construct a field with 4 elements?

Can you construct a field with 4 elements? can you help me think of any examples?
4
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1answer
160 views

Ideals and filters

The notions of a filter and an ideal on a poset make intuitive sense to me, and I can understand why they are dual: A subset $I\subset P$ of a poset $P$ is an ideal if: for all $x\in I$, $y\leq x$ ...
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3answers
153 views

Are local Artinian algebras injective?

Suppose $R$ is a local Artinian algebra. Question: Is $R$ an injective $R$-module?
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3answers
638 views

Homomorphisms from a unital ring to a ring with no zero divisors preserve unity?

I'm having a bit of trouble with a problem from Hungerford's Algebra concerning ring homomorphisms. Let $f\colon R\to S$ be a homomorphism of rings such that $f(r)\neq 0$ for some nonzero $r\in ...
4
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1answer
156 views

General form of elements in $ \mathbb{Z} [\frac{1+\sqrt{-3}}{2} ] $

What is the general form of elements in $\displaystyle \mathbb{Z} \left[\frac{1+\sqrt{-3}}{2} \right] $? I'm getting muddled. Thanks
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3answers
2k views

The subring test

This is how the wikipedia article on subring defines the subring test The subring test states that for any ring $R$, a nonempty subset of $R$ is a subring if it is closed under addition and ...
7
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1answer
568 views

Ring homomorphisms which map a unit to a unit map unity to unity?

this is the third part of a question I've been working on from Hungerford's Algebra. It is exercise 15 in the first section of Chapter III. $(c)$ If $f\colon R\to S$ is a homomorphism of rings ...
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5answers
665 views

Why is the endomorphism ring of $\mathbb{Z}\times\mathbb{Z}$ noncommutative?

So I hear that the endomorphism ring of an abelian group is not always commutative. In particular, I'm looking at the abelian group $A=\mathbb{Z}\times\mathbb{Z}$, and considering $\text{End } A$. I ...
0
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1answer
59 views

Prove a property of when simple modules are isomorphic

Let $\mathfrak{m}_1$ and $\mathfrak{m}_2$ be left maximal ideals of a unital ring $A$. Show that the simple modules $A/\mathfrak{m}_1$ and$A/\mathfrak{m}_2$ are isomorphic if and only if there exist ...
5
votes
4answers
840 views

Lack of unique factorization of ideals

I'm aware of the result that integral domains admit unique factorization of ideals iff they are Dedekind domains. It's clear that $\mathbb{Z}[\sqrt{-3}]$ is not a Dedekind domain, as it is not ...
4
votes
4answers
785 views

Proving that an ideal in a PID is maximal if and only if it is generated by an irreducible

Let $R$ be a PID (Principal Ideal Domain) and $x$ is an element R. Prove that the ideal $\langle x\rangle$ is maximal if and only if $x$ is irreducible. Ok, so I know what an irreducible is. I'm ...
4
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1answer
242 views

Lifting maps of quotient modules

Today I tried to check this, but couldn't see how to do it. I think it is probably a standard result, but a brief check of Atiyah-Macdonald didn't yield anything, and I don't know what to google for. ...
5
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1answer
147 views

Lang's “General Integrality Criterion”

Theorem 3.7 in the chapter on ring extension on page 352 of the latest edition of Lang's "Algebra" appears redundant in its phrasing to me. Specifically, if $g_s$ is a polynomial of total degree ...
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2answers
220 views

Why is $R[X,Y]/(X^2-Y^3)$ isomorphic to $\{\sum a_iT^i\in R[T] \; : \; a_1=0\}$?

Let $R$ be a ring (commutative, with unit). Show that $A=\{\sum a_iT^i\in R[T] \; : \; a_1=0\}$ is a subring of $R[T]$ and isomorphic to $R[X][Y]/(X^2-Y^3)$. Of course, I'm trying to find a ring ...
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3answers
552 views

Dedekind's theorem on the factorisation of rational primes

Let $K$ be an algebraic number field, and suppose its ring of integers is $\mathcal{O}_K = \mathbb{Z}[\theta]$ for some $\theta \in \mathcal{O}_K$. Let $f \in \mathbb{Z}[X]$ be the minimal polynomial ...
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2answers
262 views

Is there a better way of writing this ring?

The quotient $k[x,y]/(x-y^2)$ is isomorphic to $k[y]$ as a ring. Suppose, $g$ is a polynomial in $y^2$. Is there a "nice" ring that is isomorphic to $k[x,y^2]/(x^2-gy^2)$ assuming $g$ is not a unit? ...
5
votes
3answers
292 views

When is $\mathrm{Hom}(A,R) \otimes B =\mathrm{Hom}(A,B)$?

It is simple to show that if $A,B$ are vector spaces, then $A^* \otimes B = \text{Hom}(A,B)$, where $A^*$ is the dual of $A$. To what extent does this hold for modules, where we should interpret $B^*$ ...
0
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1answer
60 views

$\mathbb{C}$-vector space structure inside nontrivial f.g. module over $M_2 (\mathbb{C})$ has dim $\geq 2$

I'm trying to prove: If $M$ is a nontrivial finitely generated left module over $M_2 (\mathbb{C})$, then the accompanying $\mathbb{C}$-vector space structure (just restrict the action to the scaling ...
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5answers
1k views

Non-unital rings: a few examples

Every ring I've ever heard of is unital, i. e., contains a (unique) element $a$ such that $xa = ax = x$ for every $x$ in it. However, some rings do not have such an element. What are they? P. S.: one ...
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0answers
99 views

Geometric understanding of principal/non-principal ideals

A number field $K$ with the $r$ embeddings into $\mathbb R$ and $2s$ pairs of conjugate embeddings into $\mathbb C$ can put into ring homomorphism with the product of rings $\mathbb R^r \times \mathbb ...
2
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1answer
333 views

Methods of determining if a non-primitive polynomial is irreducible in a ring

Take, for example, $ X^4 + 2X + 2 $ in $ \mathbb{Q}[X] $. How do I determine if this is irreducible? Thoughts: I know Gauss' Lemma and Eisenstein's criterion, but they only work for primitive ...
3
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2answers
892 views

Show that if $ R $ is an integral domain then a polynomial in $ R[X] $ of degree $ d $ can have at most $ d $ roots

Show that if $ R $ is an integral domain then a polynomial in $ R[X] $ of degree $ d $ can have at most $ d $ roots. Thoughts so far: I feel like I might be missing something here. If $ R $ is an ...
3
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1answer
221 views

Let $ n \geq 3 $. By factorising $ n $ or $n + 1 $ (as appropriate), show that $ \mathbb{Z}[\sqrt{-n}] $ is not a UFD

Let $ n \geq 3 $. By factorising $ n $ or $n + 1 $ (as appropriate), shat that $ \mathbb{Z}[\sqrt{-n}] $ is not a UFD. My thoughts so far: Define $ N(a + b \sqrt{-n}) = a^2 + n b^2 $. Suppose $ n $ ...
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5answers
560 views

Exhibit an integral domain $R$ and a non-zero non-unit element of $R$ that is not a product of irreducibles.

Exhibit an integral domain $R$ and a non-zero non-unit element of $R$ that is not a product of irreducibles. My thoughts so far: I don't really have a clue. Could anyone direct me on how to think ...
4
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1answer
409 views

Give an element of $ \mathbb{Z}[\sqrt{-17}] $ that is a product of two irreducibles and also a product of three irreducibles

Give an element of $ \mathbb{Z}[\sqrt{-17}] $ that is a product of two irreducibles and also a product of three irreducibles. My thoughts so far: Using the multiplicative norm $ N(a + b\sqrt{-17}) = ...
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4answers
1k views

Show that the ideal $ (2, 1 + \sqrt{-7} ) $ in $ \mathbb{Z} [\sqrt{-7} ] $ is not principal

Show that the ideal $ I = (2, 1 + \sqrt{-7} ) $ in $ \mathbb{Z} [\sqrt{-7} ] $ is not principal. My thoughts so far: Work by contradiction. Assume that $ I $ is principal, i.e. that it is generated ...
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votes
2answers
100 views

equality of modules

I'm reading a proof of Nakayama's theorem; it says at a certain step that: For $M$, a finitely generated module on a ring $R, N$ a submodule, and $I$ an ideal of the ring $R$: If $M = N + IM$, then ...
4
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2answers
259 views

Left Artinian Rings

I have an attempt to prove the claim that $R$ has finitely many nonisomorphic simple $R$-modules if $R$ is left artinian. I would like to know if it's a good attempt. Helpful hints are very much ...
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4answers
1k views

Abstract Algebra ring homomorphism

Let $\phi\colon R \to R'$ be a ring homomorphism. Prove that if $R$ is a field then either $R$ is an isomorphism or $\phi(r) = 0$ for all $r \in R$. I am stuck on this problem and don't know ...
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1answer
65 views

Submodules of $A\times B$

I am self studying ring theory and modules from Rotman's Advanced Modern Algebra. I would like some help on putting this thought to bed. Let $A$ and $B$ be rings. Let $R=A\times B$. Is it ...
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1answer
162 views

Semisimple rings

Let R and S be rings. Show that R x S is semisimple if and only if both R and S are semisimple. I have the converse direction ($\Leftarrow$). It is the other direction ($\Rightarrow$), that I ...
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7answers
3k views

Show that $\langle 2,x \rangle$ is not a principal ideal in $\mathbb Z [x]$

Hi I don't know how to show that $\langle 2,x \rangle$ is not principal and the definition of a principal ideal is unclear to me. I need help on this, please. The ring that I am talking about is ...
4
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1answer
281 views

Some isomorphism conditions

Here I want to ask some true or false problems regarding isomorphisms (and if false, is there some extra condition to make it true). I do not what is the correct title for these problem. And I also ...
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0answers
133 views

Exterior algebras and radicals

So without giving excessive background, I was working on baby Spivak & got to the stuff about exterior algebras, & now I'm going through the section on tensor algebras in my copy of ...
15
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1answer
299 views

Does a finite ring's additive structure and the structure of its group of units determine its ring structure?

Let $A$ and $B$ be finite commutative rings with unity. Denote the additive group structure of each to be $A^{(+)}$ and $B^{(+)}$, and the multiplicative group of units of each to be $A^{(\times)}$ ...
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2answers
274 views

Minimal systems of generators for finitely generated algebras over commutative (graded) rings

Let $S$ be some base ring (a commutative ring or even just a field), and $R$ a commutative ring containing $S$ which is finitely generated (as an algebra) over $S$. What conditions guarantee that ...
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0answers
60 views

Computing relations on the columns of a matrix

Given an $m\times n$ (with $n>m)$ matrix $M$ over a polynomial ring $R=k[x_1,...,x_n]$, suppose that every column of $M$ is an $R$-linear combination of $m$ specified columns. I would like to ...
4
votes
2answers
306 views

Two different ideals with the same annihilator

Is this statment always true? $$(a)\subsetneq (b)\Rightarrow \text{Ann}_R b\subsetneq \text{Ann}_R a$$ If it is false, can you please provide an example? Also what is the largest class of rings that ...
2
votes
3answers
156 views

Simplifying expressions

I have a polynomial ring $R=k[x,y,z...]$ and a given ideal $I$ (defined by given generators) and several polynomials $f_1,f_2,...$ in the ring. I also have several other elements of $R$ given as ...
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3answers
396 views

Does this give a localization of a ring?

Apologies in advance for the naivety of this question. Let $R$ be a commutative (resp. non-commutative) ring, $S \subset R$, and let $R' = R[x_s (s \in S)]$ be the polynomial ring obtained by ...