Questions tagged [ring-theory]

This tag is for questions about rings, which are a type of algebraic structure studied in abstract algebra and algebraic number theory.

2,578 questions with no upvoted or accepted answers
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87
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Does every ring of integers sit inside a ring of integers that has a power basis?

Given a finite extension of the rationals, $K$, we know that $K=\mathbb{Q}[\alpha]$ by the primitive element theorem, so every $x \in K$ has the form $$x = a_0 + a_1 \alpha + \cdots + a_n \alpha^n,$$ ...
60
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869 views

Classification of local Artin (commutative) rings which are finite over an algebraically closed field.

A result in deformation theory states that if every morphism $Y=\operatorname{Spec}(A)\rightarrow X$ where $A$ is a local Artin ring finite over $k$ can be extended to every $Y'\supset Y$ where $Y'$ ...
23
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287 views

Semirings induced by symmetric monoidal categories with finite coproducts

A symmetric monoidal category with finite coproducts is by definition a symmetric monoidal category $(\mathcal{C},\otimes,1,\dotsc)$ such that the underlying category $\mathcal{C}$ has finite ...
12
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387 views

What properties do the rings of infinite, upper-triangular matrices have?

I'm very familiar with the ring of $n\times n$ matrices over a field, the ring of $n\times n$ upper triangular matrices over a field, and the ring of infinite column-finite matrices over a field. But ...
11
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543 views

What is the most general algebraic structure that a finite set has?

For an object $X$ in a category with finite products, define its endomorphism Lawvere theory to be the Lawvere theory generated by $X$: its $n$-ary operations are given by $\text{Hom}(X^n, X)$, and so ...
11
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181 views

Is there an elementary way to prove that the algebraic integers are a Bézout domain?

Well, the title of my question says all, but let me give some context. Right now I'm writing some lecture notes on ring theory with a little of commutative algebra. I wrote a few results about ...
11
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156 views

Can we characterize all infinite PID s whose group of units is singleton?

I am looking for a way to characterize all infinite PID s having exactly one unit i.e. invertible element ( finite PID s are not interesting , they are all fields ) . The only such example I know of ...
11
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575 views

Converse of Chinese Remainder Theorem

Chinese Remainder Theorem for commutative rings with identity Let $R$ be a commutative ring with identity. If $I, J$ are ideals of $R$ satisfying $I+J=R$, then there is an isomorphism of rings: $$R/(...
10
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1answer
132 views

On a ring $R$ such that every subring of $R$ is an ideal .

$\mathbf {The \ Problem \ is}:$ Give an example of a non-commutative ring $R$ (which may or may not contain the identity) such that every subring of $R$ is an ideal . $\mathbf {My \ approach} :$ I ...
10
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451 views

Proving a ring in which $r^n=r$ for all $r$ is commutative.

Let $R$ be a ring with identity such that there is a positive integer $n\geq 2$ for which $r^n=r$ for all $r\in R$. Prove $R$ is commutative. I had proven before that If $n=2$ it is commutative as ...
10
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1answer
645 views

Homomorphic Compression

Can there be an algorithm such that, given plaintext data P,Q, and compression function e, Such that if we treat P and Q as a number (a series of bits): $$\begin{eqnarray*}e(P + Q)& =& e(P) +...
9
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203 views

How to effectively compute fundamental units in rings?

Consider the ring of integers of $K=\mathbb{Q}(\sqrt 2,\sqrt 3)$. By Dirichlet's unit theorem the units of $\mathcal O_K$ have rank 3, so they are expressible as $\pm u_1^au_2^bu_3^c$ for suitable ...
9
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205 views

Does there exist a non-field Noetherian domain whose field of fraction is the field of real numbers?

Does there exist a Noetherian domain (but not a field) whose field of fractions is the field of real numbers $\mathbb{R}$ ? Any help will be appreciated. Thanks
9
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1answer
632 views

Applications of the Dedekind-Hasse criterion

It is a fact that an integral domain $R$ is a principal ideal domain if and only if there is a Dedekind-Hasse function $|R|\setminus\{0\}\xrightarrow{\ \ \delta\ \ }\mathbb{N}$ on $R$, i.e. a function ...
9
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295 views

Relative chinese remainder theorem and the lattice of ideals

Let $R$ be a ring with two-sided ideals $I,J$. The proof of the "absolute" chinese remainder theorem revolves around the fact that if $I,J$ cover $R$ in the lattice of ideals, i.e $I+J=R$, then the ...
9
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157 views

Classification of all subrings

Let $R$ be an integral domain whose underlying additive group is finitely generated free and whose field of fractions $K$ is a finite Galois extension of $\mathbb{Q}$. Is there a method of ...
9
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2k views

Rings with noncommutative addition

I was wondering if "rings" with noncommutative addition are studied at all? Of course, if a ring $R$ has a $1$, then for all $a, b\in R$, $a+a+b+b=(1+1)a+(1+1)b=(1+1)(a+b)=(a+b)+(a+b)=a+b+a+b$, from ...
9
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1k views

Regular Noetherian local rings are integral domains - questions about the proof

I am reading a proof that if $(A,\mathfrak m)$ is a regular local ring, then $A$ is an integral domain. I put the major questions I'm worried about in bold, but there are a lot of little things I'm ...
9
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166 views

An example of a compact multiplicatively unbounded ring

My teacher asked me to build an associative topological Hausdorff compact ring $R$ with $1$, which is multiplicatively unbounded. That means there is a neighborhood $U\ni 1$ such that $FU\not=R$ for ...
9
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201 views

Tensoring is thought as both restricting and extending?

I hope these questions are not too trivial. Let $I$ be an ideal in $R$. Write $I'\subseteq R[t]$. Then the notion of tensoring $$ (R[t]/I')\otimes_{\,\mathbb{C}[t]} \mathbb{C}[t]/\langle t-c \...
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527 views

Proof of Smith Normal Form as a Generalization of Rank-Nullity Theorem

For any matrix $A$ with entries in a PID, there exist invertible matrices $P$ and $Q$ such that $B = PAQ$, where $B$ is in Smith normal form. This theorem is usually proved by using elementary row/...
8
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143 views

Show that $n^2-1+n\sqrt{d}$ is the fundamental unit in $\mathbb{Z}[\sqrt{d}]$ for all $n\geq 3$

Let $n\in \mathbb{Z}$, $n\geq3$ and $d=n^2-2$. I want to show that $n^2-1+n\sqrt{d}$ is the fundamental unit in $\mathbb{Z}[\sqrt{d}]$. Substituting $n=3,4,5$ gives the elements $8+3\sqrt{7}$, $15+4\...
8
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0answers
72 views

Is there a non-constant $h \in \mathbb{C}[x_1 , \dots , x_n ]$ that divides every element of this given ideal?

I'm trying to prove the following: Let $I ⊆ \mathbb{C}[x_1 , \dots , x_n ]$ be an ideal with the property that any two elements $f_1$ , $f_2 \in I$ have a non-trivial common divisor. Then there ...
8
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188 views

Domains such that $R[X] \cong S[X]$ but $\mathrm{Frac}(R)[X] \not \cong \mathrm{Frac}(S)[X]$

Is it possible to find two integral domains $R,S$ such that $R[X] \cong S[X]$ but $\mathrm{Frac}(R)[X] \not \cong \mathrm{Frac}(S)[X]$ ? $\renewcommand{\Frac}{\mathrm{Frac}}$ Here $\Frac(R)$ ...
8
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241 views

The importance of being Cohen-Macaulay

I am starting to study Cohen-Macaulay rings, mainly from Bruns-Herzog book. In that book there are many examples and sentences of the type "If something satisfies this properties, then it is Cohen-...
8
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256 views

Rings satisfying “for all $a \neq 0$, there is nonunit $b$ with $a+b$ a unit”

Consider the following condition on a ring: For every nonzero $a$, there is a nonunit $b$ with the property that $a+b$ is a unit. Observe that if $a$ is already a unit, then $b=0$ will do just ...
7
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169 views

The kernel of $\mathbb{Q}[x,y]\rightarrow \mathbb{Q}(t)$ is $(x^2+y^2-1)$.

We define a ring homomorphism $F : \mathbb{Q}[x,y]\rightarrow \mathbb{Q}(t)$ by mapping $f(x,y)$ to $f(\frac{1-t^2}{1+t^2},\frac{2t}{1+t^2})$. Show $\ker F=(x^2+y^2-1)$. Proof) Since $(\frac{1-t^2}{1+...
7
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359 views

MacLane-Birkhoff's “Algebra” vs Jacobson's “Basic Algebra I,II” vs Lang's “Algebra”

Cross-posted at Math Educators Stack Exchange.. I'm searching for an apt textbook(s) on Abstract Algebra for a very advanced undergraduate/graduate level course in Algebra, and would be grateful ...
7
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1answer
78 views

$n\geq 2 $ such that the equation $x^2-x+\hat2=\hat0$ has an unique solution in $\mathbb Z_n$

Find $n\geq 2 $ such that the equation $x^2-x+\hat2=\hat0$ has an unique solution in $\mathbb Z_n$. I've tried to solve it this way: Let $a$ be its only solution. We see that $1-a$ is a solution ...
7
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0answers
148 views

When the element-wise product of two ideals produces an ideal

Consider the ring $R=\mathbb C[X,Y]$. For every two ideals $I,J$ of $R$, define $I*J:=\{ij : i\in I, j\in J\}$. Now definitely, $I*J=J*I$ always holds. If $I$ is principal, then actually $I*J$ is an ...
7
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0answers
110 views

Infinite Noetherian ring of dimension $1$ in which distinct non-zero ideals have distinct and finite index

Let $R$ be an infinite commutative ring with unity such that every non-zero ideal has finite index. Then $R$ is Noetherian, every non-zero prime ideal is maximal , and I can also show that $R$ is an ...
7
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0answers
107 views

Rings and categories with zero Grothendieck group

I am interested in examples of rings (or triangulated categories) that have zero Grothendieck group but are somehow still interesting. More example, for what rings $R$ is the category of finitely-...
7
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162 views

How to prove a specific quotient of polynomial ring is a free module?

Problem 1.19 in Eisenbud's Commutative Algebra asks the following. Given $R = k[x,y,z,w]$ and $I = (yw - z^2, xw - yz, xz - y^2)$, show that $R/I$ is free as an $k[x,w]$-module, and exhibit a ...
7
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156 views

$(x)$ prime ideal in $R[x]$ iff $R$ integral domain by contrapositive

I've done this proof a few ways and I like this one but since it wasn't the "official" one, I wanted to ask if anyone sees a reason it's invalid. It just makes more sense to me on a concrete level. ...
7
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0answers
78 views

Confusion about the definition of ideal (of ring of algebraic integers)

I am studying algebraic number theory and am confused about the following lemma. We prove that if $I \subset O_K$ a non-zero ideal then $$\textrm{disc}(I) = \textrm{disc}(O_K)\cdot N(I)$$ Then ...
7
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1answer
161 views

Commutative rings with unity over which every non-zero module has an associated prime

Let $R$ be a commutative ring with unity such that every non-zero module over $R$ has an associated prime. Then is it true that $R$ is Noetherian ? If not, then can we say something about any possible ...
7
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75 views

For what rings are all finitely generated right $R$ modules Hopfian?

In this question a user wondered if all finitely generated modules over a Dedekind finite ring (a ring satisfying $xy=1\implies yx=1$ for all $x,y$) are Hopfian (every surjective homomorphism $M\to M$ ...
7
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0answers
237 views

Determine all local rings containing $\mathbb{C}$ and having dimension 5 as $\mathbb{C}$-vector space

Let $R$ be a local ring containing $\mathbb{C}$ and $\dim_{\mathbb{C}}R = 5$ (as $\mathbb{C}$-vector space). Let $\mathfrak{m}$ be the maximal ideal of $R$ and $d=\dim_{\mathbb{C}}\mathfrak{m}/\...
7
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0answers
115 views

If prime $p=a_n10^n+a_{n-1}10^{n-1}+\ldots+a_110+a_0$ then $f(x)=a_nx^n+\ldots+a_0$ is irreducible in $\mathbb{Z}[x]$

I have been trying to solve this problem on my own for four days now, and I cannot figure out how to prove it: If we express a prime $p$ in base $10$ as $$p= a_m10^m+a_{m-1}10^{m-1}+\ldots +a_110+a_0,...
7
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0answers
412 views

What is $\Bbb Z[X]/(aX+b)$ isomorphic to?

Let $a,b$ be integers. I would to know what other ring is $R=\Bbb Z[X]/(aX+b)$ isomorphic to? If $a$ is a unit of $\Bbb Z$, then $R \cong \Bbb Z$. If $a=0$, then $R \cong (\Bbb Z/b\Bbb Z)[X]$. If $a=...
7
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0answers
195 views

Irreducibility of a family of polynomials coming from Fibonacci polynomials

Let $\{F_n(z)\,|\, n\geq 1\}$ be the Fibonacci polynomials, defined recursively by $F_1=1, F_2=z$ and $F_{n+2}=zF_{n+1}+F_n$. Now consider the polynomial $$\varphi_{n,m}(z)=4-F_n^2F_m^2(z^2+3).$$ I ...
7
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0answers
277 views

When is the polynomial $\prod\limits_{i=1}^n\,\left(x-a_i\right)+1$ reducible in $\mathbb{Z}[x]$?

This post is inspired by Prove that the polynomial $\prod\limits_{i=1}^n\,\left(x-a_i\right)-1$ is irreducible in $\mathbb{Z}[x]$.. (A) Find all positive integers $n$ and integers $a_1,a_2,\ldots,...
7
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97 views

Maximal ideals in the ring of measurable functions

The $R$ ring of continuous functions from $[0,1]$ to $\mathbb{R}$ has a property that its maximal look like a subset of $R$ consisting of those functions which vanish at a common single point in $[0,...
7
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0answers
143 views

Symmetric Algebra and Extension of Scalars

I am reading Gortz and Wedhorn's Algebraic Geometry. In their section on the symmetric algebra they explain the adjoint situation $$ \mathrm{Sym}_A \dashv i_A\colon \mathrm{Alg}(A)\to \mathrm{Mod}(A)$...
7
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0answers
113 views

Ramification group fixing an unramified extension

For a Galois extension of local fields $L/K$ with Galois group $G$, define the ramification groups $G_i$ by $$G_i = \{ \sigma \in G : \nu_{L}(\sigma(x) - x) \geq i+1 \text{ } \forall x \in \mathcal{O}...
7
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0answers
238 views

Is there a simple example of a ring that satifies the DCC on two-sided ideals, but doesn't satisfy the ACC on two-sided ideals?

It follows from the Hopkins–Levitzki theorem that if a ring satisfies the DCC on left ideals, then it also satisfies the ACC on left ideals. I've been trying to find a counterexample to the following ...
7
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0answers
496 views

Relations between semi-artinian and $\pi$-regular rings

A ring $R$ [associative, with 1, not necessary commutative] is said to be right semi-artinian if every non-zero module over $R$ has a simple submodule. A ring $R$ is said to be strongly $\pi$-regular ...
6
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0answers
165 views

Graphical multiplication tables for $\mathbb{Z}/p\mathbb{Z}$ and $\mathbb{Z}$

Inspired by Burkard Polster's beautiful video on Times Tables, Mandelbrot and the Heart of Mathematics I wondered how this graphical approach to visualize the multiplicative structure of finite rings ...
6
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0answers
278 views

Zero divisors and minimal prime ideals in commutative ring

Let $A$ be a commutative ring with unity different from $0$. Let $D(A)$ be the set of those prime ideals $\mathfrak{p}$ of $A$ which satisfy $$(*) \mbox{ there is } a\in A \mbox{ s.t }. \mathfrak{p} ...
6
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0answers
96 views

Algebra of additive polynomials

Let $\mathbb{F}_p$ be a finite field for an odd prime $p$. Consider the ring $$\mathcal{L}(\mathbb{F}_p(X),Y)$$ of additive (or linearized) polynomials in $Y$ over the rational function field $\mathbb{...