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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Integral closure of $R[x]$ in its field of fractions over R

I feel like this might have been discussed before but I couldn't find it so I apologise if this is a very common question. If $S$ is a ring and we have a subring $R$ and an element $x\in S$ ...
3
votes
1answer
19 views

Length of the primary component of $(xy, y^2)$ at the origin is $1$. [on hold]

As the question title suggests, how do I see that the length of the primary component of $(xy, y^2)$ at the origin is $1$?
0
votes
1answer
18 views

Construction of a Module isomorphism

Let $R$ be a PID. Consider the sets $X_0=\{v_0,v_1,v_2\}$ and $X_1=\{e_1,e_2,e_3\}$ and let $C_i$ be the free $R$-module on $X_i$ for $i=0,1$. Consider the $R$-module homomorphism $$C_1\;\...
5
votes
1answer
117 views

Does every finite dimensional real nil algebra admit a multiplicative basis?

We say that a finite dimensional real commutative and associative algebra $\mathcal{A}$ is nil if every element $a \in \mathcal{A}$ is nilpotent. By multiplicative basis, I mean a basis $\{ v_1, \...
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0answers
28 views

A non domain ring with every non-zero annihilating ideal a prime ideal has a particular form.

A non-domain ring in which every non-zero annihilating ideal is a prime ideal, is of the form $F_1 \bigoplus F_2$, $F_1$, $F_2$ are fields or has only one non-zero proper ideal. Note: Here, an ideal ...
0
votes
1answer
26 views

Showing a quotient ring is commutative

Question: Let R be the ring of all continuous function from R to $\mathbb{R}$ under point-wise addition and multiplication. Show that $ I=\left \{ f \in R \mid \left ( 0 \right )f=0\right \}$...
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2answers
82 views
+50

$R$ be an integral domain , $x \in R$ , $I$ an ideal such that $I+\langle x \rangle , (I:x)$ are principal ideals , then is $I$ a principal ideal?

Let $R$ be an integral domain , $x \in R$ , $I$ be an ideal such that $I+\langle x \rangle $ and $(I:x):=\{r \in R : rx \in I\}$ both are principal ideals , then is $I$ also a principal ideal ?
4
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1answer
43 views

Units in quotient rings

Let $I$ and $J$ be ideals of a ring $R=\mathbb{K}[X_1, \dots, X_m]/K$, quotient of a polynomial ring over a field $\mathbb{K}$. Consider the map $$\begin{aligned} (R/I)^*\oplus (R/J)^*&\...
3
votes
0answers
50 views

How to show that the field of fractions of a domain $A$ is $\{ab^{-1}: a \in A, b \in A-\{0\}\}$?

Let $A$ be a domain. Suppose that for all $a,b\in A-\{0\}$, $Aa \cap Ab \neq \{0\}$ and $aA \cap bA \neq \{0\}$. How to show that the field of fractions of $A$ is $\{ab^{-1}: a \in A, b \in A-\{0\}\}$...
4
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1answer
39 views

problems about prime ideals

Let $A=\mathbb{Z}[X,Y]/(Y^2-6X^2), B=\mathbb{Z}[X,T]/(T^2-6)$ where $X,Y,T $ are variables and let $x,y$ be the cosets of $X,Y$ in $A$ whilst $x',t$ be the cosets of $X,T$ in $B$. Consider the ideals ...
2
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0answers
32 views

Is the group of units in $\mathcal{O}_{\mathbb{Q}(\sqrt{2})}$ finite and cyclic?

Let $K=\mathbb{Q}(\sqrt{2})$ and let $\mathcal{O}_K$ be its ring of integers. Consider the group of units in $\mathcal{O}_K$. Is it finite? Is it cyclic? My thought: For any $\alpha = a+b\sqrt{2}$, ...
0
votes
1answer
15 views

Verify size of factor ring

Let the ring $R=\left \{ \begin{bmatrix} a_{1} &a_{2} \\ a_{3}& a_{4} \end{bmatrix} \mid a_{i} \in \mathbb{Z} \right \}$ and let I be the subset of R consisting of matrices with even ...
0
votes
1answer
36 views

What does the $(m,n)$ mean in context of $\mathbb{Z}_{(m,n)}$

On the section about the Hom sets of modules, Hungerford has an exercise that asks to show that $$\operatorname{Hom}(\mathbb{Z}_m, \mathbb{Z}_n) \cong \mathbb{Z}_{(m,n)}$$ and then in the next ...
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0answers
22 views

Unclear proof of a proposition on semisimple rings in Lang

I've been reading through Lang's Algebra chapter about semisimple modules and semisimple rings, and after Lang proves a main structure theorem about semisimple rings he proves the following ...
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0answers
24 views

$R$ is a commuttaive unital ring with no non-trivial idempotent , then is the group of units of $R$ not isomorphic with the additive group of $R$?

Let $R$ be a commutative unital ring such that $R$ has no non-trivial idempotents , let $(R^\times , .) $ denote the group of units of $R$ , then is it true that $(R,+) \ncong (R^\times ,.)$ ? See $...
2
votes
3answers
82 views

Non-principal ideal in $K[x,y]$?

Let $K$ be a field. Then $K[x,y]$ is not a PID because $x$ is irreducible but the quotient $K[x,y]/(x)$ is isomorphic to $K[y]$, which is not a field. So there must be an ideal in there which is not ...
3
votes
4answers
43 views

On the kernel of a certain module epimorphism $\mathbb{Z}^2 \to \mathbb{Z}/6\mathbb{Z}$

In order the construct a certain projective resolution of $\mathbb Z / 6 \mathbb Z$ I need to find the kernel of the ($\mathbb Z$-) module morphism: $$\epsilon_0 : \mathbb Z^2 \to \mathbb Z / 6 \...
4
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0answers
67 views

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

Does there exist a non-field Noetherian domain whose field of fraction is the field of real numbers $\mathbb{R}$ ? Any help will be appreciated. Thanks
2
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1answer
44 views

Matrices representing injective homomorphisms

Let $R$ be a ring and $M$, $N$ finitely generated free modules modules over $R$. Let $A$ be a matrix representing a homomorphism $f: M \rightarrow N$. We know that the map $f$ is injective if and only ...
1
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1answer
27 views

$R$ be an infinite commutative ring such that $R/I$ has only finitely many ideals for every non-zero ideal $I$ , what can we say about $R$?

It is known that if $R$ is an infinite commutative ring such that for every non-zero ideal $I$ , $R/I$ is finite then $R$ is a Noetheian domain . It is also known that if $R$ is a PID then for every ...
5
votes
2answers
792 views

Irreducible but not prime element

I am looking for a ring element which is irreducible but not prime. So necessarily the ring can't be a PID. My idea was to consider $R=K[x,y]$ and $x+y\in R$. This is irreducible because in any ...
4
votes
1answer
68 views

$X^2$ irreducible but not prime

Let $K$ be a field. I'd like to show that in $R=\{\sum a_i X_i\in K[X]\mid a_1=0\}$ the element $X^2$ is irreducible, but not prime. Irreducibility is checked easily, but I can't see why it's not ...
-4
votes
1answer
60 views

Set of zero divisors is an ideal iff the ring is local [on hold]

Let $R$ be a commutative ring with unity. Show that $Z(R)$, the set of all zero divisors of $R$, is an ideal if and only if $R$ is a local ring. I have no idea for proving this. Thanks in advance!
4
votes
1answer
46 views

Two questions on the Gaussian integers [duplicate]

I have two questions on the Gaussian integers. Is any element in $\mathbb{Z}[i]$ the root of a monic polynomial with coefficients in $\mathbb{Z}$? Conversely, does any element in $\mathbb{Q}(i)$ ...
4
votes
2answers
77 views

$R$ be a commutative unital ring , is it true that the group of units of $R$ is not isomorphic with the additive group of $R$?

Let $R$ be a commutative ring with unity , let $R^{\times}$ be the group of units of $R$ , then is it true that $(R,+)$ and $(R^{\times} ,\cdot)$ are not isomorphic as groups ? I know that the ...
0
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1answer
50 views

Does there exist any non-zero polynomial $f:\mathbb C \to \mathbb C$ such that $f(x+2)-2f(x+1)=f(x) , \forall x \in \mathbb C$ ?

Does there exist any non-zero polynomial $f:\mathbb C \to \mathbb C$ such that $f(x+2)-2f(x+1)=f(x) , \forall x \in \mathbb C$ ?
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1answer
45 views

Is $4x^2+3xy^2+y^3+7$ irreducible in $\mathbb{C}[x,y]$?

Is $4x^2+3xy^2+y^3+7$ irreducible in $\mathbb{C}[x,y]$? I tried to group it like $(4)x^2+(3y^2)x+(y^3+7)$. This is a polynomial with degree $2$ so I am thinking of applying quadratic formula... where ...
7
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0answers
58 views

If $A[X] \cong B[X]$ as rings, are the degrees of irreducible polynomials the same in $A$ and in $B$?

First, I ask my question and then I add some explanations: Suppose that $A$ and $B$ are two commutative rings such that $A[X] \cong B[X]$ as rings. Denote by $D_A$ the set of all positive integers ...
3
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1answer
57 views

Is there a theory of generalized eigenvectors over commutative rings?

Brown's Matrices over Commutative Rings book discusses the theory of eigenvalues, eigenvectors, and diagonalizing matrices over commutative rings, but unless I've missed something, nothing like ...
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0answers
54 views

Clifford algebra of a non-diagonal quadratic form over rings

I know how to construct explicitly the clifford algebra of a quadratic form over fields, even in the case the diagonal quadratic form over rings. But how should I $\textbf{construct explicitly}$ the ...
4
votes
2answers
174 views

Why is $S/Z$ a domain for the ideal $Z=\{z\in S\mid za=0,\;\forall a\in R\}$ in $S$?

Suppose $R$ is a rng with no zero divisors, not necessarily commutative. I know $R$ can be embedded into a ring $S:=\mathbb{Z}\times R$ by identifying $r\in R$ with $(0,r)\in S$. The operations on $S$ ...
2
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1answer
43 views

Norm of Gaussian integers: solutions to $N(a) = k$ for $k \in \mathbb{N}$ and $a$ a Guassian integer

I have two questions. Which integers are equal to the norm of some Gaussian integer? In general, how many solutions does$$\text{N}(a) = k$$have for a given $k \in \mathbb{Z}$? I am investigating the ...
4
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1answer
50 views

Quotient ring of Gaussian integers $\mathbb{Z}[i]/(a+bi)$ when $a$ and $b$ are NOT coprime

The isomorphism $\mathbb{Z}[i]/(a+bi) \cong \Bbb Z/(a^2+b^2)\Bbb Z$ is well-known, when the integers $a$ and $b$ are coprime. But what happens when they are not coprime, say $(a,b)=d>1$? — For ...
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1answer
222 views

Does $A^2 \cong B^2$ imply $A \cong B$ for rings?

If $A$ and $B$ are two unital rings such that $A \times A \cong B \times B$, as rings, does it follows that $A$ and $B$ are isomorphic (as rings)? I believe that the answer is no, but I can't come ...
3
votes
1answer
115 views

Why is ring of integers $\mathcal O_K$ called ring of integers - what properties of $\mathbb{Z}$ does it inherit?

I was wondering why ring of integers $\mathcal O_K$ for field $K$ is called ring of integers. Definition says that elements in this ring will be a solution for monic equation with coefficients of ...
0
votes
1answer
27 views

A question on extension of rings which related to their direct summands

I read "Foundations of Module and Ring Theory" of Robert Wisbauer and I got stuck in this problem: *Show for a ring $R$. The following assertions are equivalent: (a) $R$ has a unit. (b) If $R$ is ...
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2answers
106 views

Nilpotent or non-Nilpotent Jacobson Radical

Let $R$ be a ring with identity element such that every ideal of which is idempotent or nilpotent. Is it true that the Jacobson radical $J(R)$ of $R$ is nilpotent? If $R$ is Noetherian and $J(R)$ is ...
5
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1answer
1k views

Correspondence theorem for rings.

Could someone provide a reference that includes a full and honest proof of the Correspondence Theorem for rings? Let $A$ be a multiplicative ring with identity and $I$ an ideal of $A$. There is a ...
0
votes
3answers
26 views

Minimal ideal in a ring which is generated by an idempotent element.

Let $R$ be a commutative ring with unity and $M$ be a minimal ideal of $R$ such that $M = Re$ where $e$ is an idempotent element in $R$. Then $R = Re \oplus R(1-e) $ I am not able to see, in order ...
3
votes
3answers
502 views

Showing that the elements $6$ and $2+2\sqrt{5}$ in $\mathbb{Z}[\sqrt{5}]$ have no gcd

In showing that the elements $6$ and $2+2\sqrt{5}$ in $\mathbb{Z}[\sqrt{5}]$ have no gcd, I was thinking of trying the following method. If the ideal $(6)$ + $(2+2\sqrt{5})$ is not principal in $\...
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0answers
35 views

Algebra of Endomorphisms of a functor $\mathcal{F}$

Recently, I was reading a paper and stumbled upon something for which I don't find the formal definition unfortunately in any textbook that have in my hands. It's the notion of the endomorphism ...
2
votes
2answers
46 views

What are the conditions needed for two principal ideals of a ring to be isomorphic?

Given a commutative ring $R$, and $p(x),q(x) \in R[x]$ monic polynomials, under what conditions on $p(x)$ and $q(x)$ are the principal ideals $\langle p(x) \rangle$ and $\langle q(x) \rangle$ ...
2
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1answer
43 views

$\Bbb Z[\sqrt{-5}]$ is not a PID [duplicate]

I want to show: In a PID $R$ two elements $a,b\in R$ always have a greatest common divisor. Therefore $\Bbb Z[\sqrt{-5}]$ is not a PID. For the first part: $I=\{ax+by:x,y\in R\}$ is an ideal, ...
0
votes
1answer
29 views

Is $\Bbb Z[i]$ a Euclidean ring? [duplicate]

Is $\Bbb Z[i]$ a Euclidean ring? If not, what would be the simplest way of seeing that $\Bbb Z[i]$ is a PID?
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1answer
25 views

Given a commutative ring $R$ and a monic polynomial $p(x) \in R[x]$ is $R[x]/\langle p(x) \rangle$ always a finite integral extension of $R$?

I suspect this to be true based on the fact that $p(x)$ is monic, so it should be the case that $R[x]/\langle p(x) \rangle$ is a finitely generated module over $R$, but I have no good reference for ...
1
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2answers
43 views

Left- and right-sided principal ideals have same index?

One fact about the Lipschitz integers (quaternions of the form $a + bi + cj + dk$ where $a, b, c, d$ are integers) is that the left-sided ideal generated by any element $Q$ has the same index in the ...
1
vote
1answer
35 views

Proving that a prime ideal $p \subset R$ yields a prime ideal $p[x] \subset R[x]$

I'm curious as to whether I can have my proof critiqued. Proposition : Let $\mathfrak{p}$ be a prime ideal in a ring $R$. Show that $\mathfrak{p}[x]$ is a prime ideal in $R[x]$. Proof : Suppose $\...
1
vote
2answers
50 views

Prove that if $I$ is maximal, then $R[X]$ is a PID. [duplicate]

Let $R$ be a commutative ring with unity such that $R[X]$ is a UFD. Denote the ideal $\langle X\rangle $ by $I$. Prove that If $I$ is maximal, then $R[X]$ is a PID. If $R[X]$ is a Euclidean Domain ...
0
votes
2answers
36 views

$R/Rg$ is a field iff $g\in R$ is irreducible.

Let $R$ be a PID and $g\in R$. I want to show: $R/Rg$ is a field iff $g\in R$ is irreducible. I.e. I want to show that all $a\notin Rg$ are invertible modulo $g$ iff $g$ is irreducible. So if I ...
2
votes
1answer
29 views

Bijection beteween maximal ideals

We know that if $R$ and $I$ an ideal of $R$, then there is a bijection between the prime ideals of $R$ containing $I$ and the prime ideals of $R/I$. It is given by $P\mapsto P/I$. Is it true that this ...