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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Why are principal fractional ideal also fractional ideals?

I don't understand the following: Let $K$ be the quotient field of an integral domain $R$. A fractional ideal $I$ is a subset of $K$ other than $\{0\}$, for which a $0\neq r\in R$ exists, so that ...
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2answers
66 views

Show that the rings $2\mathbb{Z}$ and $3\mathbb{Z}$ are not isomorphic.

Here I am under the impression that $2\mathbb Z$ and $3\mathbb Z$ are the sets of even numbers and multiples of $3$ respectively and the operations are usual addition and multiplication. This is an ...
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1answer
44 views

Equivalence between category of $R$-modules and $S$-modules

Is it true that by equivalence between category of $R$-modules and $S$-modules we conclude that $R$ and $S$ are isomorphic?($R$ and $S$ are unital rings.)
2
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1answer
27 views

Arbitrary elements in a quotient ring $\Bbb R[x]/(x-1)$

If I have an ideal $(x-1)$ for the ring $\Bbb R[x]$, how do I think of the quotient ring $\Bbb R[x]/(x-1)$? I have all polynomials with: $$a_nx^n+a_{n-1}x^{n-1}+\cdots+a_2x^2+a_1x+a_0 {\pmod {x-1}}$$ ...
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1answer
35 views

about left identity in a ring..

Let $S$ be the subset of $\mathbb{M}(\mathbb{R})$ consisting if all matrices of the form : $\begin{pmatrix} a & a \\ b & b \end{pmatrix}$ The matrix $\begin{pmatrix} x & x \\ y & y ...
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0answers
39 views

Ideals in a quotient

I am interested in finding the number of prime ideals in $\mathbb{Z}[x]/(12,x^2+1)$. Here is what I think. Modding out by $x^2+1$, we get $\mathbb{Z}[i]/(12)$. Factoring $12$ in the Gaussian integers, ...
3
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2answers
46 views

Understanding Quotient Rings

I am watching a video on Field Extensions (trying to self "relearn" some Abstract Algebra before I take it again). I struggled with it as an undergrad, so I'm trying to get a leg up. The example ...
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2answers
35 views

How to go about this proof for non zero polynomials.

How do I go about proving this? Let $\mathbb{F}$ be a field and $X$ an indeterminate, and consider the polynomial ring $\mathbb{F}[X]$. Let $f(X), g(X) \in \mathbb{F}[X]$ with $f(X), g(X) \neq ...
2
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1answer
37 views

Zeroes of prime polynomials in the algebraic torus (A Hilbert's Nullstellensatz for Laurent polynomials?)

Let $Q\in\mathbb C[z_1,\dots,z_D]$ be a prime polynomial and let $Z(Q)$ be the algebraic hypersurface of its zeroes. Assume that $P\in\mathbb C[z_1,\dots,z_D]$ is a polynomial which has zeroes at ...
2
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1answer
35 views

Finite covering by finitely generated B-algebras

While I was working on the first properties of schemes on Hartshorne's Books, I needed the following result that I couldn't prove it: Let $X=Spec(A)$ be an affine scheme. Suppose that there exist an ...
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1answer
39 views

Algebraic closure for rings

Is there any notion of algebraic closure for commutative rings? I am specifically interested in such a concept for $\mathbb Z_n$, with $n$ not a prime (possibly square-free). Such a concept would be ...
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2answers
50 views

prove $(m) \subset (n)$ iif $n$ divides $m$

For non-zero integers $m$ and $n$, prove $(m) \subset (n)$ iif $n$ divides $m$, where $(n)$ is the principal ideal. My attempt is following. For non-zero integers $m$ and $n$, assume that $(m) ...
2
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2answers
56 views

Proving $0x=0$ in a ring

I am trying to prove the above trivial statement. I am aware of the standard approach of letting $0 = 0 + 0$ and cancelling, but I would like the below statement to be verified/corrected: $1\cdot ...
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1answer
26 views

Polynomial Rings

I know that an Euclidean ring is a principal ring and this last is a factorial ring. I also know that if $B$ is a field, then $B[x]$ is Euclidean. While, for instance $\mathbb{Z}[x]$ is not a ...
2
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1answer
33 views

About the ways prove that a ring is a UFD.

I'm doing an exercise where I have a commutative ring with unity $R$. We had to find that the nonunits formed an ideal (maximal). After that, we found the irreducible elements, and then we saw that ...
2
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1answer
33 views

If some non-primitive polynomial in $\mathbb Z[x]$ is irreducible over $\mathbb Q$, does this imply it is irreducible over $\mathbb Z$?

I know from a lemma in Herstein that if a primitive polynomial in $R[x]$ is irreducible in its field of quotients $F[x]$, then it is irreducible in $R[x]$. But, if some non-primitive polynomial in ...
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3answers
726 views

Are there more groups than rings?

It seems pretty clear to me that both of these are at least uncountable (which I think I could prove with some work). It also seems that you should be able to make some diagonal argument about the ...
0
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1answer
46 views

What does the ring $R=C[x]/I$ look like?

Maybe it's a stupid question but what does the ring $R=C[x]/I$ look like? $I$ is the ideal in $C[x]$. Everything helping! Thanks :)
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1answer
118 views

DVR, power series expansion.

Let $A$ be a discrete valuation ring with quotient field $K$, maximal ideal $\mathfrak{m}$, uniformizing parameter $t$. Let $k = A/\mathfrak{m}$, so $k$ is a field. How do I show that there is a ...
5
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1answer
46 views

Only DVR's with quotient field $\mathbb{Q}$?

Let $p \in \mathbb{Z}$ be a prime number. I know how to show that $$\{r \in \mathbb{Q}: r = {a\over{b}},\text{ }a,b \in \mathbb{Z},\text{ }p\text{ doesn't divide }b\}$$ is a DVR with quotient field ...
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1answer
26 views

Local ring coincides with DVR.

Assume $A$ is a discrete valuation ring with quotient field $K$ and maximal ideal $\mathfrak{m}$. If $S$ is a local ring containing $A$ and contained in $K$ with maximal ideal containing ...
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2answers
38 views

A question on Join homomorphism and Ideals

On page 287 of the book Mathematical Methods in Linguistcs, by Barbara Partee, Alice Ter Meulen and Robert E. Wall (Dordrecht, Kluwer Academic Press, 1993), I find the following theorem, which they ...
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0answers
81 views

Regular subrings of a polynomial ring

Let $R=\mathbb{C}[x,y]$. I have the following situation: $\mathbb{C} \subseteq D \subseteq R$ is affine (=finitely generated as a $\mathbb{C}$-algebra), noetherian, has field of fractions ...
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42 views
+100

Example of a commutative, local, dual ring with nilradical $N$ such that $ann(N)\subsetneq N$

For an ideal $I\lhd R$ in a commutative ring $R$, let $ann(I)$ denote the annihilator of $\{x\in R\mid xI=\{0\}\}$. A commutative ring $R$ is said to be a dual ring if for every ideal $I$ of $R$, ...
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1answer
57 views

$R$ is Finite ring and for every $a \in R$, there exist natural number $n(a)$ ST $a^{n(a)}=a$

$R$ is Finite ring and for every $a\in\,R$ there exist natural number $n(a)>1$ that $a^{n(a)}=a$ . Is $R$ is a ring with identity? If this question is correct then, for every $a \in ...
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0answers
18 views

If $I$ is a finitely generated semisimple left $R-$module, show that $I=Re$ for an idempotent $e \in I$.

Let $I$ be a left ideal in a semiprime ring $R$. If $I$ is a finitely generated semisimple left $R-$module, show that $I=Re$ for an idempotent $e \in I$.
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1answer
32 views

Regularity of a quotient ring of the polynomial ring in three indeterminates

Let $I=(f)$ be a prime ideal in $R=\mathbb{C}[x,y,z]$, so $f$ is an irreducible polynomial, and further assume that $f$ is of the following form: $f=z^n+c_{n-1}z^{n-1}+\ldots+c_1z+c_0$, where ...
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1answer
80 views

Question about ideals of a ring: $I\cdot J=I \implies J=I$?

Doing exercises, this question came to my mind. Is it true that if $I$ and $J$ are proper and nonzero ideals of a ring $R$, $$I\cdot J=I \implies J=I?$$ And $$I\cdot J=I \iff J\subseteq I?$$
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Properties of a localization of $\mathbb Z$

Let $R=\{\frac ab ∈ ℚ ∣ b \text{ is odd}\}$. (1) Prove that $R$ is isomorphic to $ℤ_P$, where $ℤ_P$ is the localization at $P$, for a prime ideal $P$ of $R$. (ii) Find $U(R)$. Prove that ...
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2answers
60 views

Ideals of the localization of a ring [on hold]

Let $R$ be a domain. Let $P$ be a prime ideal of $R$. (i) Prove that $S:=R\setminus P$ is a multiplicatively closed system with no zero divisors. Prove that $RP=S^{−1}R=\{a/b∈K\mid b∉P\}$. (ii) We ...
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1answer
47 views

Ring morphism is injective or not [duplicate]

I've got an exercise here that states the following: Let $R$ be a commutative ring and $S$ be a subset of $R$, with $S$ multiplicatively closed. If $T$ is a commutative ring and $φ'\colon R \to T$ ...
2
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1answer
20 views

Describing the Kernel of an identity-preserving ring morphism from a ring $R$ to an Endomorphism ring of an additive Abelian Group.

I'm currently working through TS Blyth's book on Module Theory (Module Theory: An Approach to Linear Algebra). From Exercise 2.3: "Let $M$ be an Abelian Additive Group, and $R$ a unitary ring. Let ...
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2answers
64 views

Problem about ideals of the localization of a ring

I'm having problems on doing the section (ii) of this exercise. Let $R$ be a domain. Let $P$ be a prime ideal of $R$. (i) Prove that $S:=R\setminus P$ is a multiplicatively closed system with no ...
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0answers
78 views

Matrix representation of $\mathbb{C}$ as $^*$Algebra.

We know that there are many matrix representations of the field $\mathbb{C}$. For $2 \times 2$ real entries matrices, e.g., all the subrings of $M(2,\mathbb{R})$ generated by $I$ and a matrix $J$ such ...
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1answer
41 views

Isomorphism of polynomial rings [closed]

How I prove, $\mathbb{R}/(x^2)$ is isomorphic to Dual number Ring. Dual Number ring The dual number ring is define by $D={a+bϵ:a,b∈R,ϵ^2 =0}$
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1answer
45 views

Number of generators of prime ideals in $K[x_1,x_2,…,x_n]$

Is there any bound for the number of generators of prime ideals in $K[x_1,x_2,...,x_n]$? (For example in $K[x,y]$.) We know that maximal ideals of $K[x_1,x_2,...,x_n]$ have $n$ generators.
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1answer
34 views

$K[x,y]$ (where $K$ is a field) have any bound for the number of generators of ideals?

We know that maximal ideals of $K[x_1,x_2,...,x_n]$ have $n$ generators. But is there any bound for the number of generators of arbitrary ideals? (For example in $K[x,y]$.)
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2answers
117 views

Let $R=\{a/b \in \mathbb{Q} \mid \text{$b$ is odd}\}$

Let $R=\{a/b \in \mathbb{Q} \mid \text{$b$ is odd}\}$. Prove that $R$ is isomorphic to $\mathbb{Z}_P$, where $\mathbb{Z}_P$ is the localization at $P$, for a prime ideal $P$ of $R$. I really ...
3
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1answer
61 views

Nilpotents after tensoring with a field

Let $A \to B$ be a homomorphism of commutative rings with unit. Let $A_{\text{red}}=A/ \sqrt{(0)}$ and $B_{\text{red}}=B/ \sqrt{(0)}$ be the corresponding reduced rings. Now let $A_{\text{red}} \to K$ ...
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3answers
53 views

In a commutative ring with identity, if $p$ is irreducible, is ($p$) a maximal ideal?

In a Euclidean Domain, $D$, if we mod out by an irreducible, $p$, we get the field $D/(p)$. I can see that this follows since we are going to be able to write $1$ as a linear combination of $p$ and ...
3
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2answers
49 views

If $1\in M$, could $M$ be a maximal ideal of a commutative ring with identity $R$?

If $1\in M$, could $M$ be a maximal ideal of a commutative ring with identity $R$? I know that this is a very silly question. I think that the answer is that $M$ can't contain the identity for ...
3
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1answer
52 views

Working in $\mathbb Q[x]$. Two polynomials are coprime if their gcd is a constant?

When are two polynomials coprime? Is it when their gcd is a constant? If we divide $x^3-7x-5$ by $x-4$, we get: $$x^3-7x-5=(x-4)(x^2+4x+9)+31$$ So, is $31$ their gcd, but since $31$ is not monic ...
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1answer
46 views

Injectivity of composed homomorphisms. [on hold]

I'm doing this exercise: Let $R$ be a commutative ring and $S$ be a subset of $R$, with $S$ multiplicatively closed (that is: $1 \in S$, $ab\in S$ if $a,b\in S$ , $0$ is not in $S$ and $S$ has not ...
3
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2answers
48 views

about center of group rings $RG$ and $(R/I)G$

Let $I$ be an ideal of a ring $R$. It is mentioned in the book An Introduction to Group Rings (by Sehgal and Milies) that the canonical homomorphism $RG \rightarrow (R/I)G$ maps $Z(RG)$, center of ...
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0answers
74 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 ...
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0answers
45 views

Evaluating composition of functors

Let $R$ be a ring and $S$ its $n \times n$ matrix ring. We consider the categories $_R Q$ and $_S Q$ of their respective left modules. We define a functor $F \colon _R M \to _S M$ by $$ F(M) = M^n $$ ...
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1answer
59 views

How to show $1+\sqrt 2$ generate an infinite cyclic group of units in $\mathbb Z[\sqrt 2]$?

The answers given here seem very convoluted: The units of $\mathbb Z[\sqrt{2}]$. Is it possible to provide a more explanatory proof?
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4answers
358 views

infinitely many ideals

does the ring $\Bbb Z_2[x]$ have infinitely many ideals like $\Bbb Z[x]$? How do you know if a ring has a finite number of ideal. particularly asking about seemingly large rings.
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0answers
9 views

finding prime elements in Z[√2 ] [duplicate]

I am trying to find prime elements of z[√2],and I'am trying to have a procedure like finding prime elements in z[i],is it correct or not?? tnx for your help.
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1answer
58 views

Is $\mathbb{Q}(\sqrt{3})$ in someway related to Quotient ring?

I can't help but notice that they look exactly the same. For example: $\mathbb{Q}(\sqrt{3})$ = $\lbrace p + q\sqrt{3}:p,q \in \mathbb{Q}\rbrace$ That seems pretty much exactly an ideal. Only the ...