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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5
votes
1answer
78 views

Algebraic closure of the rational inside a quotient of product of finite fields

I'm trying to solve the following exercise: " Consider the ring $R = \prod_{p} \mathbb{F}_p$, where $p$ runs over all prime numbers and $\mathbb{F}_p$ is a field with $p$ elements. Show that there ...
1
vote
1answer
32 views

what inequalities can one have between $depth\ R$ and $depth\ M$? when $depth\ R \geq depth\ M$

Let $(R,m)$ be a commutative Noetherian local ring which is not CM. Let $M$ be a finite $R$-module. what inequalities can one have between $depth\ R$ and $depth\ M$? Obviously there are ...
1
vote
3answers
72 views

A question about the relation between division ring and domain

"Is it true that any division ring is a domain?" Note 1: I am not sure "domain"="integer domain", are they different? Note 2: Since the definition of integral domain, I can't see if a division ring ...
0
votes
1answer
82 views

Fiber as vector space over residue field.

Let $A$ be a commutative ring with identity and let $M$ be an $A$-module. The fiber of $M$ at $P \in \text{Spec}A$ is the module $M(P):=M_P / PM_P$, which is a vector space over the residue field ...
1
vote
1answer
22 views

Integral closure of a DVR in finite extension of fraction field

Let $(K,|\cdot|)$ be a complete valued field and let $L$ be a field extension with $[L:K]<\infty$. Let $\mathcal{O}_K$ be the valuation ring in $K$ and let $\mathcal{O}_L$ be the integral closure ...
4
votes
2answers
90 views

A question on Mumford's drawing of $\text{Spec}\,\mathbb{Z}[x]$

This might seem like a really silly question, but what are those weird curves connecting $(x^2 + 1)$ and $(5, x+2)$ in Mumford's picture of $\text{Spec}\,\mathbb{Z}[x]$?
1
vote
1answer
48 views

Near-rings: why are ideals defined like that?

While I am learning the very basics of Near-rings ,I have come across the following statement: " a subgroup I of N is an ideal iff $$n \equiv m \pmod I$$ and $$x \equiv y \pmod I$$ implies ...
0
votes
1answer
24 views

Example where $R = F[A]$ is an integral domain.

Let $A\in M_n(F)$ be an $n \times n$ matrix over the field $F$ such that $A$ is not a scalar multiple of the identity. Let $R = F[A]$. Find examples where $R$ is an integral domain, and one where ...
2
votes
1answer
31 views

What can the differential operator map to?

Let $W$ be the Weyl algebra $\mathbb{C}[x, \partial_x]$, that is the span of two elements $x, \partial_x$ with $$ \partial_x x - x\partial_x = 1. $$ If $\phi$ is a ring automorphism $\phi : W \to W ...
-4
votes
1answer
72 views

Example: $ u\in A \otimes_R B$, but $u \neq a \otimes b $ for any $ a \in A, b \in B.$ [closed]

Give an example to show the following may actually occur for suitable ring $R$ and modules $A,B$: $ u\in A \otimes_R B$, but $u \neq a \otimes b $ for any $ a \in A, b \in B.$
2
votes
1answer
68 views

Endomorphisms of the ring $(\mathbb{Z}/p\mathbb{Z})^n$

Describe the set of Endormorphisms of the ring $ K = (\mathbb{Z}/p\mathbb{Z})^n$ where $p$ is prime. I think my main difficulty is viewing $K$ as ring versus viewing it as a ...
2
votes
1answer
93 views

Show that the ring $R$ is Noetherian [closed]

Let $$R = \{ \frac{f(z)}{g(z)}: f, g \in \mathbb{C}[z], g(z) \neq 0 \text{ for } |z| = 1 \}.$$ Prove that $R$ is a Noetherian ring. Note : $\mathbb C$ is the set of the complex numbers, $z$ is a ...
0
votes
1answer
42 views

example of an ideal $I$ in an integral domain $A$ for which there is a prime in $\text{Ass}(A/I)$ that is not in $\text{Ass}(A)$

What is an example of an ideal $I$ in an integral domain $A$ for which there is a prime in $\text{Ass}(A/I)$ that is not in $\text{Ass}(A)$? I've tried constructing one, but all my attempts have ...
3
votes
0answers
45 views

Finite Ring with unity and no zero divisors is field

I would like to know if someone can help me with this. "Show that a finite ring $R$ with unity $1\neq 0$ and no divisors of 0 is a field." The original exercise asked me to show that it was a ...
1
vote
0answers
27 views

Field of polynomials mod n?

I have a few questions and i am looking for some clarification. 1) Is it correct that one can define a field $(Z_n, +, X)$ of integers mod $n$, where all the elements are integers $a$ such that ...
0
votes
1answer
51 views

Calculate the support of module

Let $A=k[x,y]$ where $k$ is an algebraically closed field and let $M=A/(xy)$ be an $A$-module. I am supposed to calculate $\text{Supp}(M)= \{ P \in \text{Spec}(A) : M_p \not= 0 \}$ where $M_p = ...
3
votes
4answers
48 views

Is the complement of a prime ideal closed under both addition and multiplication?

Let $P$ be a prime ideal in a commutative ring $R$ and let $S=R-P$ ,i.e. $S$ is the complement of $P$ in $R$. Then, justify with reason which of the following(s) are correct: $S$ is closed under ...
2
votes
1answer
21 views

Equivalent conditions between two ideals and a nilpotent ideal in a ring. [duplicate]

Let $R$ be a ring. Prove that for any two ideals $I$ and $J$ of $R$, the following conditions are equivalent: $(1)$ $IJ = (0) \implies I\cap J = (0) $; $(2)$ $(0)$ is the only nilpotent ...
0
votes
3answers
88 views

Find the field of fractions and the integral closure of a subring of $\mathbb Z[x]$.

Let $R$ be a subring of $\mathbb{Z}[x]$ consisting of polynomials such that the coefficients of $x$ and $x^2$ are zero. Find the field of fractions of $R$. Find the integral closure of $R$ in it's ...
3
votes
0answers
32 views

cohomology of complex grassmannians and quaternion grassmannians (the finite dimensional case)

Let $G_k(\mathbb{C}^N)$ be the complex grassmannian manifold. Let $G_k(\mathbb{H}^N)$ be the quaternion grassmannian manifold. Let $p$ be a prime integer (we may also impose extra conditions, such as ...
3
votes
1answer
156 views

Splitting of an exact sequence

Let $(R,\mathfrak m)$ be a Noetherian local ring. Suppose that $x \in \mathfrak m \setminus \mathfrak m^2$. Is it true that $$ \frac{\mathfrak m}{x\mathfrak m} \cong \frac{\mathfrak m}{(x)} ...
3
votes
1answer
32 views

Every nilpotent ideal is a nil ideal.

Let $R$ be a ring. An ideal $I$ of $R$ is called nilpotent if $I^{n} = {0}$ for some positive integer $n$. I want to show that every nilpotent ideal is a nil ideal. Please help me.... i can't start ...
2
votes
0answers
65 views

If every maximal ideal of $R$ is principal, is every ideal of $R$ principal? [duplicate]

Let $R$ be a commutative ring and every ideal maximal of $R$ is principal (generated by only one element). Is every ideal of $R$ principal? Please help me my friends. It's necessary for me.
0
votes
1answer
76 views

Find the ring of homomorphisms $\mathbb{Z} \to \mathbb{Z}$

Show, which well-known ring is isomorphic to ring $End(\mathbf{Z})(+,ͦ,-,0,id)$ of homomorphisms $\mathbf{Z}$ -> $\mathbf{Z}$, where $\mathbf{Z}$ is commutative group $\mathbf{Z}(+)$ and 0 constant ...
0
votes
2answers
61 views

Is $\{x f(x)+3g(x) \;|\;f,g\in \mathbb{Q}[x]\}$ a (main) ideal?

Is it possible to show whether or not $ \{xf(x)+3g(x)\;|\;f(x),g(x) \in \mathbb{Q}[x]\} $ is an ideal (or main ideal in $\mathbb{Q}[x]$)? I know how to prove it for $\mathbb{Z}[x]$, but what with ...
6
votes
1answer
74 views

Boolean algebra gives rise to a ring

There is a well-known result that every boolean algebra $(R,0,1,\land,\lor,\neg)$ can be made a boolean ring by defining $$x\cdot y=x\land y\qquad\text{and}\qquad x+y=(x\land\neg y)\lor(y\land\neg ...
2
votes
1answer
65 views

Integral Domain but not a UFD

Let $R = \mathbb{Z}[x,y]$. Find ideals $I$ such that. $R/I$ is an integral domain but not a UFD The polynomial $z^2 - 1$ has more than two roots in $R/I$. For 1, I have $I = (x^2 - xy -1)$ which ...
1
vote
1answer
42 views

How to conclude that whether a given polynomial is irreducible or not?

For which $n\in \{2,3,7\}$ is the polynomial $x^3+x^2+x+2$ irreducible in $\mathbb{Z}/(n)[x]$? My work: For a given ring $R$ and a polynomial $p(x)\in R[x]$, if $p(\alpha)=0$ for some $\alpha\in ...
5
votes
1answer
125 views

Proving a subring of $\mathbb{Q}$ containing $\mathbb{Z}$ is a PID

Let $S$ be a subring of $\mathbb{Q}$ containing $\mathbb{Z}$. Prove that it is a principal ideal domain. So here is what I tried. Take any ideal $I\subset S$. Take any two elements, say $a=p/q, ...
2
votes
1answer
56 views

Find all the maximal ideals in the ring $\mathbb{R}[x]$.

Find all the maximal ideals in the ring $\mathbb{R}[x]$. My work: If $M$ is maximal, then $\mathbb{R}[x]/M$ is a field. Then if $M$ is of the form $(p(x))$ then $p(x)$ must be irreducible. So ...
3
votes
1answer
59 views

Centralizer of semisimple matrix

Let $n>3$ be an integer. I have trouble finding a semisimple matrix $A\in M_n(\mathbb R)$ (i.e diagonalizable in $M_n(\mathbb C)$) such that its centralizer $Cent(A)=\{X\in M_n(\mathbb ...
3
votes
0answers
28 views

For which countable successor ordinals $\alpha$ is the reverse order isomorphic to the ideals of a PID ordered by inclusion?

Let $\alpha$ be a countable successor ordinal and $\alpha^{\mathrm{op}}$ the reverse order. For which $\alpha$ is there a commutative principal ideal ring $R$ such that the ideals of $R$ form a chain ...
4
votes
3answers
86 views

$\mathbb{Z}[X]/(2x+4,x^2-3) \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ [duplicate]

Someone has asked a question before regarding the ring $\mathbb{Z}[X]/(2x+4,x^2-3)$, but the answer wasn't quite what I was looking for. I was wondering how one would show this quotient isomorphic to ...
1
vote
2answers
59 views

Why would a field *not* be considered a discrete valuation ring?

There are two theorems in Matsumura (p. 78-9) Theorem 11.1 Let $R$ be a valuation ring. Then the following conditions are equivalent: (1) $R$ is a DVR (2) $R$ is a PID (3) $R$ is ...
1
vote
2answers
41 views

Show that the rings $R_1=F_5[x]/(x^2)$ and $R_2=F_5\times F_5$ are not isomorphic.

Show that the rings $R_1=F_5[x]/(x^2)$ and $R_2=F_5\times F_5$ are not isomorphic.($F_5$ is the field with $5$ elements.) My Work: Since $(0,1)$ does not have an inverse, $F_5\times F_5$ is not a ...
0
votes
1answer
117 views

Prove that some canonical homomorphism is injective.

Let $A \not= \{0 \}$ be a Noetherian commutative ring and let $M$ be an $A$-module. Prove that the canonical homomorphism $$M \to \bigoplus_{P \in \text{Ass}(M)} M_p$$ is injective. My question is, ...
1
vote
1answer
33 views

Representation of Algebraic Extensions by Matrices

Let $\mathbb C$ be the field of complex numbers and $\mathbb R$ the field of real numbers. It is well known that the field $\mathbb C$ can be represented as $$\mathbb ...
3
votes
0answers
68 views

Can $\operatorname{Spec}(A)$ be expressed as an inverse limit?

We know that given a ring $A$ such that $A/\mathfrak{R}$ is absolutely flat, then $\operatorname{Spec}(A)$ is Hausdorff (it's an equivalence). So $Spec(A)$ becomes a quasi-compact, Hausdorff and ...
1
vote
1answer
135 views

How to find algebraic connections between zeros of a polynomial?

Let $f(x)$ be an irreducible integer polynomial of degree $k$. Let $x_1,x_2,...,x_j$ be some zeros of $f(x)=0$ where $j<k$. How do I find identities of type $P(x_1,x_2,...,x_j) = 0$ where $P$ is ...
0
votes
1answer
33 views

Any finite integral domain is a field [duplicate]

I'm going through the Rings section of Abstract Algebra by Dummit and Foote, and I have a question about an early Corollary's proof. Proof: Let R be a finite integral domain and let a be a nonzero ...
0
votes
1answer
79 views

Zero dimensional Gorenstein ring

Let $(R,\mathfrak m)$ be a zero dimensional Gorenstein ring and $\mathfrak q$ be an $\mathfrak m$-primary ideal of $R$. Then TFAE: 1) $\mathfrak q$ is irreducible, 2) $(0:\mathfrak q)$ is principal, ...
0
votes
1answer
88 views

meadows and fields, aren't $0^{-1}=0$ can be proven simply from the axioms of fields?

Recall field axioms In this article http://www-compsci.swan.ac.uk/~csjvt/JVTPublications/RationalsAsADT.pdf Page 4, we have the SIP \begin{matrix} \left(-x\right)^{-1}=-\left(x^{-1}\right) \\ ...
2
votes
1answer
48 views

A nonprincipal ideal and a nonprime irreducible in $\mathbb{Z}[\sqrt{-17}]$

The problem asks to find a nonprincipal ideal and a nonprime irreducible in $R = \mathbb{Z}[\sqrt{-17}]$. Since $-17 \equiv 3 \pmod 4$, $R$ is the ring of integers of $\mathbb{Q}(\sqrt{-17})$. I ...
1
vote
4answers
116 views

Find a “simpler description” for $\mathbb{Z}[X]/(X-5,X^2+3)$

The problem asks for a "simpler description" of the ring $\mathbb{Z}[X]/(X-5,X^2+3)$. I could use the Chinese Remainder Theorem if $\mathbb{Z}$ were replaced by $\mathbb{Q}$, but here the ideals ...
3
votes
3answers
81 views

Can the complex numbers be realized as a quotient ring?

Can the complex numbers be realised as some $R/M$ where $R$ a ring and $M$ a maximal ideal like the integers modulo some prime? I understand that unlike the latter case, such a maximal ideal would ...
4
votes
2answers
78 views

finitely generated subgroup of $\mathbb{Q}^n$

For such a seemingly standard problem, I can't seem to find a reference for it... Prove that if $A$ is a finitely generated subgroup of $\mathbb{Q}^n$ then it has the form $\{\sum_{i=1}^k n_ia_i:n_i ...
0
votes
1answer
40 views

Tensor product of quotient and kernel

In my problem I have a PID $R$, elements $0\neq a,b\in R$ and a map $\phi_a:R\rightarrow R$ where $r\mapsto ar$. Assuming I have done all my previous calculations right I need to prove that ...
0
votes
2answers
102 views

$\mathbb{Z}\bigl[1+\sqrt{-7}\bigr]$ is not Euclidean

The similarly looking ring $\mathbb{Z}\Bigl[\dfrac{1 + \sqrt{-7}}{2}\Bigr]$ actually is Euclidean (see here). Now I want to show that $\mathbb{Z}\bigl[1 + \sqrt{-7}\bigr]$ is not Euclidean. ...
1
vote
0answers
50 views

$\mathbb{Z}[(1 + \sqrt{-7})/2]$ is euclidean

Show that $\mathbb{Z}\Bigl[\dfrac{1 + \sqrt{-7}}{2}\Bigr]$ is a Euclidean ring. Ok, there are some hints here but not a full proof. My attempt so far: Proof: Define $\omega := \dfrac{1 + ...
0
votes
0answers
16 views

Element in distinct prime ideal

Let $K$ be a number field and $\mathcal O$ its ring of algebraic integers. Let $\mathfrak p_1,\ldots , \mathfrak p_n$ prime ideals of $\mathcal O$ with $\mathfrak p_i\neq\mathfrak p_j$ if $i\neq j$. ...