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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322
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15k views

A short proof for $\dim(R[T])=\dim(R)+1$?

If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$. For noetherian $R$, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and non-trivial ...
31
votes
0answers
329 views

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 $$x = a_0 + a_1 \alpha + \cdots + a_n \alpha^n$$ ...
28
votes
0answers
489 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 $Y'$ ...
22
votes
0answers
367 views

Ring structure on the Galois group of a finite field

Let $F$ be a finite field. There is an isomorphism of topological groups $(\mathrm{Gal}(\overline{F}/F),\circ) \cong (\widehat{\mathbb{Z}},+)$. It follows that the Galois group carries the structure ...
18
votes
0answers
202 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 ...
13
votes
0answers
153 views

On the order of $\mathbb{Z}[X]/(f,g)$ and the resultant $R(f,g)$.

I suspect that $\#\mathbb{Z}[X]/(f,g)=|R(f,g)|$ holds for any two non-constant polynomials $f,g\in\mathbb{Z}[X]$, where $R(f,g)$ is the resultant of $f$ and $g$. I am however unable to prove it. I'd ...
12
votes
0answers
194 views

Ideals in $C[0,1]$ which are not finitely generated (From Atiyah- Macdonald )

I'm trying to solve the following problem from Atiyah-Macdonald: Is the ring of continuous function on $[0,1]$ is Noetherian ? Certainly not, here are two non terminating ascending chain of ...
11
votes
0answers
132 views

Study of rings of the form $R+I$

In my life I saw lots of ways of constructing rings: polynomial rings, quotient rings, localizations, endomorphism rings, rings of fractions, integral closure of a ring, center of a ring, etc... These ...
9
votes
0answers
222 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/(...
9
votes
0answers
112 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
votes
0answers
203 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 ...
9
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0answers
126 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
votes
0answers
180 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 \...
8
votes
0answers
81 views

Looking for a special class of ideals such that If every ascending chain of ideals from this class stabilizes, then $R$ is a Noetherian ring.

A commutative ring $R$ is called Noetherian if any one of the following holds: $1.$ Every ascending chain of ideals in $R$ stabilizes, that is, $$ I_1\subseteq I_2\subseteq I_3\subseteq\cdots $$ ...
8
votes
0answers
217 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
votes
0answers
158 views

The difference between the ring version and module version of Chinese Remainder Thereom.

Chinese Remainder Theorem for Commutative Rings If $R$ is a commutative ring with $1$ and $I, J$ are ideals of $R$ that are pairwise coprime or comaximal (meaning $I + J = R$), then $IJ = I \cap J$, ...
7
votes
0answers
330 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) +...
7
votes
0answers
473 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 ...
7
votes
0answers
99 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}...
6
votes
0answers
40 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,...
6
votes
0answers
142 views

Can we characterize all infinite Euclidean-domains having exactly one invertible element?

$\mathbb Z_2$ and $\mathbb Z_2[x]$ are two euclidean-domains having exactly one invertible element ; my question is ; Can we characterize all euclidean domains $D$ having exactly one invertible ...
6
votes
0answers
138 views

An example of prime ideal $P$ in an integral domain such that $\bigcap_{n=1}^{\infty}P^n$ is not prime

I am looking for an example of prime ideal $P$ in an integral domain such that the ideal $\bigcap_{n=1}^{\infty}P^n$ is not a prime ideal. This is a followup to this question where the ring was not ...
6
votes
0answers
215 views

Reducing multivariate rational fractions to lowest terms

I wish to simplify multivariate rational fractions to a canonical form. Thanks to some very helpful mathematically inclined people who verified that my understanding of Wikipedia was correct, I'm now ...
6
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0answers
87 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)$...
6
votes
0answers
104 views

Functorial approach to Ideals and Quotients, Multiplicative Sets and Localizations

I have been playing with substructures of commutative rings today and noticed that there is a strong analogy between the formation of quotients and kernels with the formation of localizations with ...
6
votes
0answers
134 views

Isomorphism between finite fields adjoining a root

Let $p(x)=x^3+x^2+1$ and $q(x)=x^3+x+1$ be polynomials over the field $\mathbb{Z}_2$. Let $\alpha$ be a root of $p(x)$ and $\beta$ be a root of $q(x)$. Now let $K=\mathbb{Z}_2(\alpha)$ and $F=\mathbb{...
6
votes
0answers
94 views

When are all ring homomorphisms also algebra homomorphisms?

Let $k$ be an algebraically closed field, and let $A,B$ be two unitary $k$-algebras. In general, there are more ring homomorphisms $A\to B$ than there are $k$-algebra homomorphisms. More precisely, ...
6
votes
0answers
192 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 ...
6
votes
0answers
141 views

Is there an upper bound to the number of rings that can be obtained from a semigroup with zero by defining an additive operation?

Let $\mathscr S$ be the class of all semigroups with zero. For $(S,\times,0)\in\mathscr S,$ I want to count additive operations $+$ on $S$ such that $(S,+,\times,0)$ is a ring (possibly without unity)....
6
votes
0answers
350 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/column ...
5
votes
0answers
139 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 ...
5
votes
0answers
103 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 ...
5
votes
0answers
119 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 ...
5
votes
0answers
74 views

Prove $\sqrt{-7} \not\in \mathbb{Z}\left[\frac{2+3\sqrt{-7}}{4}\right]$

I have the following problem. Consider the ring $\mathbb{Z}$ and define: $$x = \sqrt{-7}\qquad z = \frac{2+3x}{4}$$ Show that $\mathbb{Z}[x] \not\subset \mathbb{Z}[z]$ and $\mathbb{Z}[z] \not\...
5
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0answers
147 views

Showing $\sqrt{2} \notin E.$

Let $E$ denote the least subring of $\mathbb{R}$ that is closed under the operation $r \mapsto e^r$. Then presumably, $\sqrt{2} \notin E.$ Question. How can we show this?
5
votes
0answers
201 views

Counterexamples for lcm-gcd identity and modular law for rings

In Miles Reid's Undergraduate Commutative Algebra, Exercise 1.3, we need to find counterexamples of lcm-gcd identity and modular law in the ring $A=k[X,Y]/(XY)$: $(I+J)(I\cap J)=IJ$; $I\cap(J+K)=(I\...
5
votes
0answers
164 views

$A$ a ring, ($\exists n\in\mathbb{N},n\geq 2,\ \forall x\in A$, $x^n=x$) $\Rightarrow$ A is commutative?

Let $A$ be a ring. Suppose that there exists $n\in\mathbb{N},n\geq 2$ such that $\forall x\in A,x^n=x$. Does that imply that $A$ is commutative ? I have read all the links suggested in the comments, ...
5
votes
0answers
58 views

Is there an online database somewhere that lists identities for algebraic structures with two binary operators?

I'm working on an abstract algebra library in Python, and I'm trying to include as many functions that analyze algebraic structures, returning true or false based on whether or not the algebra ...
5
votes
0answers
174 views

A few questions about a specific ring

My question is kinda long, so please bear with me... And I only need hints to get me started. So, I'm working on the ring $R =\left( \begin{matrix} \mathbb{Z} & \mathbb{Q} \\ 0 & \...
5
votes
0answers
142 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-...
5
votes
0answers
199 views

“Evaluation Homomorphisms” for Formal Power Series

In the ring of formal power series $\Bbb R[[x]]$ it is easy to check by induction that $$ 1 = (1-x)(1 + x + x^2 + \cdots). $$ Does this derivation imply the same identity for those real or complex ...
5
votes
0answers
271 views

Ideals in Gaussian integers

Let $R:=\mathbb{Z}[i]$. Prove that every nonzero prime ideal $\mathfrak{P}$ of $R$ belongs to one of the following families: 1) $\mathfrak{P}=(1+i)R$ 2) $\mathfrak{P}=(a+bi)R$ where $a,b\in\mathbb{Z}...
5
votes
0answers
610 views

Integral homomorphism induces a closed map on spectra

I'm trying to prove the following: Let $f:A\rightarrow B$ be an integral homomorphism (e.g. $B/f(A)$ is a integral extension). Consider $f^{*}: \operatorname{Spec}B \rightarrow \operatorname{Spec}...
5
votes
0answers
104 views

Deciding whether or not a class of modules is “big enough”

For the last few days I'm pondering the following question. The situation is this: $R$ is a commutative ring and $A$ a (noncommutative) $R$-algebra. I have a class $\mathcal{C}\subseteq\coprod_{S} SA-\...
5
votes
0answers
234 views

System of polynomial equations over rational field

Fix $n\geq 2$. Let $p:=x_1^2+\ldots+x_{n-1}^2+1\in\mathbb{Q}[x_1,\ldots,x_{n-1}]$. Suppose $u_1,\ldots,u_n,v_1,\ldots,v_n\in\mathbb{Q}[x_1,\ldots,x_{n-1}]$ satisfy the following equations: $u_1v_1+\...
5
votes
0answers
161 views

An algebraic algorithm for finding inverses in the group algebra

This is an extension to my earlier question. Is there a purely algebraic algorithm to find inverses in the group algebra? For example, in the group algebra $\mathbb{C}S_{4}$, how would one go about ...
5
votes
0answers
468 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 ...
4
votes
0answers
43 views

Left ideals of $M_2(K)$ with $K$ a field

Is it true that the only proper left ideals of $M_2(K)$, the ring of the matrices whose coefficients are in a field $K$, are $$ \left\{\begin{pmatrix}ah & ak \\ bh & bk\end{pmatrix}: a,b \in K\...
4
votes
0answers
80 views

How do ring theorists think about square roots?

Let $R$ denote a commutative ring. Then it seems to me that we can adjoin to $R$ a square-root of $4$ as follows: $$R[\sqrt{4}] = R[x]/(x^2-4)$$ This defines a functor $\mathbf{CRing} \rightarrow \...
4
votes
0answers
78 views

What was the genesis of Hua's identity?

Many resources I have read prove Hua's identity more-or-less mechanically. I have seen there is more than one raison d'être for Hua's identity: e.g. its connection to the fundamental theorem of ...