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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Ring Integration

In thinking about various methods of integration, I began to wonder if there was some sort of unifying theory relating integration and ring theory. For example, would there be a way to make sense of ...
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Rings and Categories of Modules by Anderson and Fuller: Corollary $7.4$

I am reading the book Rings and Categories of Modules by Anderson and Fuller. I don't understand corollary $7.4$ of that book. Can anyone explain that corollary to me? Thank for any insight.
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Units and zero divisors in $\mathbb{Z_2[x]}/(x^2 + 1)$ and in general

The elements of $\mathbb{Z_2[x]}/(x^2 + 1)$ are polynomials in $\mathbb{Z_2[x]}$with degree $0$ or $1$. I.e. $\{\overline{0}, \overline{1}, \overline{x}, \overline{x+1}\}$ $\overline{0} = ...
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homework: rings, matrices and polynomials

$A,B$ are both $n \times n$ and diagonal matrices. Prove that there is a matrix $X$ which is $n \times n$, and polynomials $p$ and $q$ such that $A= p(X), B= q(X)$ Is this true for ANY 2 matrices (we ...
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Minimal spectrum of graded rings

Let $R$ be a left Noetherian ($\mathbb N$-)graded ring and let $R_0$ be its $0$-th component. When $R$ is commutative it is well-known, and easy to prove, that the minimal prime ideals of $R$ are ...
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The integral closure of Bezout domains in arbitrary field extensions is Prüfer?

Let $R$ be a Bezout domain with quotient field $K$, $L$ an arbitrary extension field of $K$, and $\overline{R}$ the integral closure of $R$ in $L$. Is $\overline{R}$ a Prüfer domain? If the answer is ...
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Help about trace of a module

I'm reading about Trace and Reject, which definition: $Tr_M(U)= \sum \lbrace Im(h)|h\in Hom_R(U,M) \rbrace$ $Rej_M(U)= \bigcap \lbrace Ker(h)|h \in Hom_R(M,U)\rbrace$ So is that right: if ...
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Quotient of complete linearly topologized ring

The quotient of a complete metrizable group by a closed normal subgroup is always complete, but there are examples to show this need not be true for non-metrizable groups. Here complete means every ...
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determine the roots of $X^{2}-1$ in $\left(\mathbb{Z}/36\mathbb{Z}\right)[X]$

I have no clue what to do here. Any tips/hints? I already got 1 and -1=35 Because $(-a)^{2}-1=(a)^{2}-1$ i only need to check the first 18 digits. And because $(2k)^2 - 36 \cdot q \neq 1$ the even ...
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Semirings (or Rigs)

Can anyone give me some famous quotients of semirings or Rigs (i.e. semirings described by generators and relations)? For example M. Fiore and T. Leinster have given the normal form for the elements ...
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Computing ext over graded rings

This question came up as I was reading Beilinson, Ginzburg, Soergel paper Koszul Duality Patterns in Representation Theory. Suppose that $A$ is a Koszul ring (for the definition of Koszul ring ...
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Product of ideals closed set

Let $R$ be a topological ring (in fact $R$ is metrizable) and let $I,J$ denote ideals of $R$. Suppose also that they are closed with respect the topology of $R$. Is it always true that the product ...
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If $\mathbb{C}[G]$ is Noetherian and $G$ has a representation on $V$, when must $V$ be finite-dimensional?

I know this is a bit vague, but please bare with me here. Let's assume that $G$ is a finitely-generated torsion group. I want to show that $G$ is a finite group if I add some conditions. I suspect ...
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Kronecker's approach to unique factorisation in algebraic number theory: books and references

I have done a short course (one semester) on algebraic number theory at beginning graduate level, in which the Dedekind theory of ideals features prominently. However I have since discovered that ...
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A question about the consequence of Prime Avoidance.

I have found the following statement: Let $R$ be a Noetherian ring and $x$ is a non-zero divisor of $R$. Let $P$ be a prime ideal associated to $xR$. Then by Prime Avoidance there exists a non-zero ...
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A criterion for finite generation of subalgebras of a polynomial ring

In her 1926 paper on invariant theory, Emmy Noether uses a certain "finiteness criterion" which I wish to translate and maybe find a more modern reference to. The original wording is: Ein ...
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Functors that have a natural Isomorphism

Find different functors $T, S: Rng \rightarrow Rng$ both identity on objects IE: for each ring $R, T(R)=S(R)=R$, such that there is a natural isomorphism between T and S. I know that a natural ...
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In $ℤ/Nℤ$, which units are successors to zero divisors?

What are the units $x$ in $ℤ/Nℤ$ of the form $x = 1 + \overline{kd}$ for a divisor $d$ of $N$ and $k ∈ ℤ$, i.e. $$U_N[d] := \{x ∈ (ℤ/Nℤ)^×;\; ∃ k ∈ ℤ : x = 1 + \overline{kd}\} = \ker \big((ℤ/Nℤ)^× → ...
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Krull dimension of a module over the Weyl algebra

Let $A_{n}$ be the $n$th Weyl algebra over a field, with generators $x_{1}, \ldots, x_{n}, y_{1}, \ldots, y_{n}$. Why is the Krull dimension of the rigth $A_{n}$-module $A_{n}/x_{1}A_{n}$ equal to ...
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Prove that $[L,Rad(L)] \subseteq N$ for finite-dimensional Lie algebra $L$

I need to prove the following fact: if $L$ is a finite-dimensional Lie algebra over field of characteristic $0$, $Rad(L)$ is its radical, and $N$ is the maximal nilpotent ideal in $L$, then ...
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Frobenius from Hurwitz's theorem

Can we deduce Frobenius theorem from Hurwitz's theorem on Normed division algebra? Frobenius theorem states that the only associative finite dimensional division algebras over the real numbers are R, ...
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A commutative ring with alternating and commutativity properties with infinite distinct elements

Is there any nontrivial commutative ring without multiplicative identity that satisfies alternating property ($x \cdot x = 0$ for all $x$ where $\cdot$ is multiplication operator and $x \cdot y \neq ...
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verifying if an ideal is prime

Let $R=\mathbb{Z}[\sqrt{-5}]$ and let $\mathfrak{i}=(2,1+\sqrt{-5})$ the ideal generated in $R$ by $2$ and $1+\sqrt{-5}$. I want to prove that $\mathfrak{i}$ is prime. So i considered the surjective ...
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Why $\text{top}h$ is an isomorphism?

I am reading the book Elements of representation theory of associative algebras I have a question about from Line -9 to Line -6 of page 29, the proof of Theorem 5.8. How to show that ...
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Left ideals of central simple algebra generated by symmetric element

Let $F$ be a field of characteristic $\neq 2$. Suppose $(A,\sigma)$ is central simple algebra over a field $F$ with an orthogonal involution. Assume $(A,\sigma)$ is non-split. How to show that every ...
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A necessary and sufficent condition for a ring to be a UFD

I came across the following question in Hungerford's Algebra: An integral domain $R$ is a UFD iff every non-zero prime ideal contains a nonzero prime principal ideal. The forward direction is ...
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Non-commutative integral extensions?

In Commutative algebra there is a notion of an integral extension: Let $P$ be a subring of $R$. Then $R$ is the integral extension of $P$ if each element of $R$ is a root of a monic polynomial with ...
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Multiplication structure for finite abelian rings of order $p^2$.

Consider an odd prime $p$ and a positive integer $a$ as close to 2 as possible that is not a quadratic residue $mod$ $p$. If we extend the ring $mod$ $p$ with the element $b = a^{\frac{1}{2}}$ then ...
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Field of Fractions for Commutative Ring with Identity

I'm trying to complete a problem that is walking me through creating a field of fractions $F$ from a commutative ring with identity (and no non-zero divisors) $R$. The construction first asks me to ...
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Projective dimension of simple module

Let $R$ be a commutative ring and $M$ a simple $R$-module. Then $\mathfrak{m}=Ann(M)$ is a maximal ideal of $R$. Then it is known that $$ \mathrm{pdim}_{R}(M)=\mathrm{pdim}_{R_{\mathfrak{m}}}(M), $$ ...
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Writing out an isomorphism between two rings

I am having a hard time writing a bijective map between the two rings: $$ R = \dfrac{k[x,y,z,u,v]}{\left<(x-y)z+uv\right>} \cong \dfrac{k[x,y,z,u,v]}{\left<(x-y)z\right>} = S. $$ I ...
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$k[V]^G = \widetilde{A}$ where $\widetilde{A}$ is the normalization of $A$

Let $V$ be a finite dimensional vector space over $k =\overline{k}$ and let $G$ be a subgroup of $GL(V)$ so that $k[V]^G$ is finitely generated. Let $A$ be a subring of $k[V]^G$ that is finitely ...
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What are the prime ideals of $\mathbb{R}[x_1,x_2,x_3,…]$

What are the prime ideals of $\mathbb{R}[x_1,x_2,x_3,...]$? (this is the ring of polynomials over the reals with countably infinite many indeteminates). My attempt: I think taking the principal ...
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Degree 1 elements in a graded ring from a blow-up perspective

This may be an elementary question but I hope this question will benefit others as much as myself. Let $k^4 = Spec \; k[x_1, x_2, x_3, x_4]$. Writing $Bl_{\mathcal{I}}(k^4)$ as $R = ...
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Simple $R$-module where $R$ is a semisimple ring. Possible small improvement of a proof.

Reading through the proof of the following theorem (in Introduction to Group Rings, by Milies and Sehgal) Let $L$ be a minimal left ideal of a semisimple ring $R$ and let $M$ be a simple ...
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Solving linear inequalities over rings

The concrete problem: for any given $N\ge 1$ I have a system of $2^N-1$ linear inequalities over $\mathbb{Z}_6^N$ which looks like this: for every nonempty $S\subseteq[N]$ there is some ...
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What does it mean for a subset of $\mathbb{C}[x_1,\dots,x_n]$ to be algebraically independent?

What does it mean for a subset of $\mathbb{C}[x_1,\dots,x_n]$ to be algebraically independent? Particularly I'd like to know the formulation thereof which concerns the kernel of a surjective ring ...
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Why is the $\mathbb{Z}$-span of a set of representations an ideal of the representation ring?

I am studying a proof of Brauer's theorem. The proof makes use of the following claim, which I haven't been able to convince myself of: Let $G$ be a finite group and let $R[G]$ be the representation ...
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Proving that a map is an isomorphism from an essential extension to itself.

I have a commutative ring $R$, and a prime ideal $P$ of $R$. I also have a module $E$ such that $R/P$ is a submodule of $E$ and every submodule $F\neq (0)$ of $E$ satisfies $F\cap R/P\neq (0)$. ($E$ ...
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Linearly independent rows in a square matrix

Suppose $m,n \in \mathbf{N}, m\le n$. Let $A$ be a matrix with $\mathbf{Q}$ linearly independent $b_{1},...,b_{m}$ in $\mathbf{Z}^{n}$. a) Show that there are $v_{1},...,v_{m} \in \mathbf{N}$ so ...
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A question about reduced torsion abelian groups

If a reduced torsion abelian group has no cyclic direct summands of order greater than 2, is it an elementary abelian 2-group? Background: I'm trying to classify the groups whose group rings have a ...
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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 ...
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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 ...
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What is it called when a subalgebra contains its centralizer?

In the question Math.SE #16716, Natalia asked about representing rings of matrices as centralizers of a matrix. This is an intriguing question, but had some clear problems as rings of matrices need ...
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Uniqueness of the decomposition of an ideal

Let $ F $ be a non-empty subset of $ \{ 1,2,\dots,n\} $ and $ P_{F}=(\{x_{i}:i\in F\}) $. Let $ F_{1},F_{2},\dots,F_{m} $ be pair-wise distinct non-empty subsets of $ \{1,2,...,n\} $ and $$ ...
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Left Ideals and Two-Sided Ideals of Rings of Matrices

This is a question I posted in this topic: Pathologies in "rng". However, I decided that the question deserves its own thread, and I want to know if anybody can answer it. For ...
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Smallest Two-Sided Nearring

For those who are unfamilar with nearrings, here is a definition. A two-sided nearring is a triplet $(N,+,\cdot)$, where $(N,+)$ is a group and $(N,\cdot)$ is a semigroup such that we have both left ...
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Computing injective hulls over a lower triangular matrix ring

Let $R$ be the ring $\begin{pmatrix} {\mathbb Z }_{ p } & 0 \\ {\mathbb Z }_{ p } & {\mathbb Z }_{ p } \end{pmatrix}$ with $p$ prime. $R$ is an artinian ring and is also a $\mathbb ...
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Generalized fact in ring theory about irreducible elements

It is quite easy to show that for $A$ an integral domain, an element $a \in A$ is irreducible if and only if the principal ideal $\langle a \rangle$ is maximal for inclusion among proper principal ...
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homology of mapping telescope of a monoid

Let $M$ be a monoid with multiplication $\cdot$, $\pi_0(M)=\mathbb{N}$, and $m\in M$ in the component $1\in \mathbb{N}$ . We form a mapping telescope $$ M\overset{ {\cdot m}}\longrightarrow M\overset{ ...