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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2answers
59 views

Is $R$ an integral domain?

Let $R = \{a + b\alpha |\ a,b \in \mathbb{Z}\}\subseteq \mathbb{C}$ where $\alpha = \frac{1}{2}(1+\sqrt{-19})$ Is $R$ an integral domain? To show whether or not $R$ is an integral domain, letting $x ...
2
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2answers
305 views

Prime implies irreducible

In a unique factorization ring with unity (I am not considering commutativity and zero divisors in definition of UFD) irreducible implies prime. And it was proved in ring with unity without zero ...
0
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4answers
344 views

Identify quotient ring $\mathbb{R}[x]/(x^2-k), k>0$

I need to identify $\mathbb{R}[x]/(x^2-k)$, where $k>0$ (if $k<0$ I believe it's isomorphic to $\mathbb{C}$). If we let $f(x) = x^2-k$, then according to Artin, since $\sqrt{k}$ satisfies $f(x)=...
0
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1answer
58 views

Compute the distance between two elements in a ring

Given a ring of size $n = 2^m$, starting with element $0$ to element $n-1$, what general formula gives the distance between two arbitrary elements $i$ and $j$? Note that the distance between the ...
3
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1answer
75 views

Find a non-principal ideal in $ \Bbb Z [2i]$.

Find a non-principal ideal in $ \Bbb Z [2i]$. I think it might be $(1+2i,1-2i)$, but have problems with proving this. I know that $|1+2i|=|1-2i|=5$. Moreover, there are only 6 elements with non-...
0
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3answers
293 views

Find prime ideals of the ring $\Bbb Z [ \sqrt[3]2]$ which contain $5$

Find all prime ideals $p$ of the ring $ R= \Bbb Z [ \sqrt[3]2] $ such that $ 5 \in p$ and find $R/p$ for all of them. I know that of course $R$ is prime and $R/R = \{0 \}$. Unfortunately I have no ...
0
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1answer
61 views

Ring isomorphism

What is the simplest form of $\mathbb{Z}[X]/(X^2+2X)$ ? I tried to use the first isomorphism theorem, but I have problems if finding a proper map $\phi$ such that $\ker\phi=(X^2+2X)$.
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1answer
147 views

Cardinality of the ring $F_3[x] / (x^2-x+1)$

How would I find the number of elements of the ring $F_3[x] / (x^2-x+1)$? I know that $x^2-x+1$ is not prime/irreducible, since gcd($x^2-x+1$, $x^3-x^2-1$) = 3. Can anyone provide some tips?
2
votes
1answer
114 views

Example of a compact module which is not finitely generated

Let $R$ be a ring and $M$ be an $R$-module. Definition: $M$ is called compact if $\text{Hom}_R(M,-)$ commutes with direct sums, that is, if for any set $I$ and any $I$-indexed family of $R$-...
6
votes
1answer
109 views

LCM of Polynomials

I know for integers we have $$\operatorname{lcm}(n,m) = \frac{nm}{\gcd(n,m)}$$ Does this hold for polynomials? i.e. $\operatorname{lcm}(f(x),g(x)) = \dfrac{f(x)g(x)}{\gcd(f(x),g(x))}$
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2answers
75 views

Which of these rings are isomorphic

Which of these rings are isomorphic: 1) $ \Bbb Z [ X] / (X^2+2X) $ 2) $ \Bbb Z [ X] / (X^2 -2) $ 3) $ \Bbb Z [ X] / (X^2+4X + 2) $ 4) $ \Bbb Z [ X] / (X^2) $ 5) $R= \{ (a,b) \in \Bbb Z ^2 \mid a \...
0
votes
1answer
32 views

Prove that a set is a vector space and a ring.

Let $$\begin{array}{ccc} M : & \mathbb{R}^3 & \longrightarrow & M_3(\mathbb{R}) \\ & (a, b, c) & \longmapsto & {\begin{pmatrix} a & b & c \\ b & a+c & b \\ ...
1
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1answer
88 views

Invariant dimension property and a ring epimorphism

In Hungerford's Algebra, p. 186, the Proposition 2.11 says Let $f:R\to S$ be a nonzero epimorphism of rings with identity. If $S$ has the invariant dimension property (IDP), then so does $R$. It is ...
0
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1answer
158 views

UFD, prime and Irreducible

I am taking following definitions and calling algebraic structure U1 and U2 definition as: U1 is A ring R with unity and properties properties Every element of R is neither 0 nor a unit can be ...
0
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1answer
76 views

Must $1+X$ be irreducible in any polynomial ring $R[X]$?

Let $R$ be a commutative ring with a multiplicative id. The question reads: Let $r$ be an element of a ring $R$. Show that, in the polynomial ring $R[X]$, the polynomial $1+rX$ is a unit if and ...
0
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1answer
90 views

Prime element in ring without unity

Definitions of prime element: $(1)$ We say $p$ is prime if $p|ab$ it implies $p|a$ or $p|b$ (I don't need definition of unity here) $(2)$ We say $p$ is prime if $p=ab$ it implies $p|a$ or $p|b$ (I ...
0
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1answer
74 views

Prime which is not irreducible in non-commutative ring with unity without zero divisors

In a non-commutative ring with unity without zero divisors find a prime element which is not irreducible (if possible). $p$ is prime iff $p|ab$ implies that $p|a$ or $p|b$, and $x$ is irreducible ...
2
votes
3answers
46 views

Prove that $(−1_R)a = −a$ (edited with new attempt)

Let R be a ring with unity $1_{R} ∈ R$ Prove that $(−1_R)a = −a$. This is done like this: $(−1_R)a + a = (−1_R)a + 1_Ra = ((−1_R) + 1_R)a = 0_R*a = 0_R$. In short, we have, $(−1_R)a + a = 0_R$, ...
3
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1answer
106 views

Non-commutative ring (not necessarily with multiplicative identity) of order $n$ exists if and only if $p^2|n$ for some prime $p$?

Is it true that there is a non-commutative ring (not necessarily with unity) of order $n$ if and only if $p^2\mid n$ for some prime $p$ ?
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2answers
81 views

How are $2\mathbb{Z}\ncong3\mathbb{Z}$ different as rings?

How are $2\mathbb{Z}\ncong3\mathbb{Z}$ different as rings? What interesting properties does one have that the other doesn't?
2
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1answer
31 views

$I=(I:s)\cap (I, s)$

Somewhere I've read the following: Theorem Let $I \subset A$ an ideal of a domain $A$. Let $S$ a multiplicatively closed set and let be $I^e$ the image of $I$ in $S^{-1}A$. Let $s \in S$ be such ...
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2answers
75 views

How to show that if $\alpha \in \mathbb{Z}[\sqrt{2}]$ and $\alpha$ is a unit, then we cannot have $1 < \alpha < 1 + \sqrt{2}$.

How can I show that if $\alpha \in \mathbb{Z}[\sqrt{2}]$ and $\alpha$ is a unit, then we cannot have $1 < \alpha < 1 + \sqrt{2}$. Assuming $\alpha = a + b\sqrt{2}$ is a unit, $1 < \alpha <...
1
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2answers
194 views

Showing that $R[X]/(Xf-1) \cong R[1/f]$ [duplicate]

Let $R$ be an integral domain with quotient field $K$. Let $0 \neq f \in R$. I want to prove Statement: $R[X]/(Xf-1) \cong R[1/f]$. Argument: Consider the epimorphism $\phi: R[X] \rightarrow R[1/f]...
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2answers
31 views

How to establish isomorphism between these quotient and product rings?

My problem: Let $A$ and $B$ be two rings, let $I$ be an ideal of $A$ and $J$ an ideal of $B.$ Prove that $I \times J$ is an ideal of $A \times B$ and $\dfrac{A \times B}{I \times J} \cong \dfrac{A}{I} ...
0
votes
1answer
38 views

isomorphism between $k[[x]]$ into $\varprojlim_n k[x]/(x^n)$ [duplicate]

i want to find isomorphism between $k[[x]]$ and $\varprojlim_n k[x]/(x^n)$ but I cant.please help me to find this.
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1answer
43 views

$v \times w$ is a bilinear map, antisymmetic and $u \times w =0 \Leftrightarrow $ collinear in tensor product

This is my Attempt for part (b): Let's define: $$\Phi: \mathbb{R}^2 \times \mathbb{R}^2 \longrightarrow \mathbb{R}^2 \otimes \mathbb{R}^2 $$ by the following action: $$\Phi(v \times w) = v \otimes w -...
4
votes
1answer
201 views

Formal power series ring, norm. [closed]

Let $k$ be a field. Let $R$ be the formal power series ring $k[[x]]$. Define $N$ on $R \setminus \{0\}$ as follows: $N(f)$ is the smallest $n$ of which the coefficient of $x^n$ in $f$ is nonzero. (a) ...
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0answers
20 views

Dimension of the vector space given by the quotient of an Artin ring by the product of all its maximal ideals [duplicate]

Let $\mathcal{M}_1,\dots,\mathcal{M}_r$ be all the maximal ideals of an Artin ring $A$ which is a finite $\mathbb{K}$-algebra; so let $A/\mathcal{M}_1\cdots\mathcal{M}_r$ be a $\mathbb{K}$-vector ...
1
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1answer
64 views

All possible zero divisors of the ring $\Bbb Z_n \oplus \Bbb Z_m$.

What can be the all possible zero divisors of the ring $\Bbb Z_n \oplus \Bbb Z_m$, where $n$ and $m$ belongs to $\Bbb N$?? One can easily verify that $(\bar a,\bar0)$ and $(\bar 0 , \bar b)$ are zero ...
3
votes
3answers
43 views

Verifying that a given set is an ideal

I am trying to show that I have said: f=0 is in I so I is non-empty. Let $f,g$ be members of $I$ $(f+g)(\sqrt5)=f(\sqrt5)+g(\sqrt5)=0+0=0 ==> f+g ∈ I$ so I is closed under addition Let $g∈Q[x],...
2
votes
1answer
45 views

$F$ is isomorphic to $\Bbb Z_p$ for some prime number $p$. [duplicate]

Suppose $F$ is a field and there is a ring homomorphism from $\Bbb Z$ onto $F$. Then show that $F$ is isomorphic to $\Bbb Z_p$ for some prime number $p$. I am facing difficulty to do the proof. I ...
0
votes
1answer
35 views

Ideals and quotient rings [duplicate]

I am trying to show that (R/I)/(J/I) is isomorphic to R/J I and J are both ideals of the ring R, and I is a subset of J. How do I begin this proof?
3
votes
1answer
60 views

Are these topologies equivalent?

Consider the space $(\mathbb C^2 , τ_1 )$ where $τ_1$ is the product topology on $\mathbb C^2$ with $\mathbb C$ having the Zariski topology i.e. closed sets indexed by $p(x) \in \mathbb C[x]$ are ...
1
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1answer
43 views

Proving if $ \Gamma_{2}(R)\smallsetminus J(R) $ is a forest then it is either totally disconnected or a star graph

These days I am reading the research paper Graphs associated to co-maximal ideals of commutative rings by Hsin-Ju Wang. In this paper, $ R $ denotes a commutative ring with the identity element. $ \...
1
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1answer
78 views

Does taking quotients preserve isomorphism of rings?

Let $R$ be a commutative ring and $A$ and $B$ be subrings of $R$. Suppose also that an ideal $I$ of $R$ is contained in both $A$ and $B$ (so $I$ is an ideal of both $A$ and $B$). I have two questions:...
2
votes
1answer
53 views

Do there exist such sets in Spec(Z[x])

Consider the topological space $(Spec(\mathbb Z[x]), τ )$ where open sets $D_I$ in τ are given as(indexed by ideals I in $\mathbb Z[x]$): $$D_I = \{p \in Spec(\mathbb Z[x])|I \not\subset p\}$$ Let $...
0
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1answer
32 views

Example of a Dedekind Finite Ring Which is Not Stably Finite

I know that there is a Dedekind Finite Ring which is not Stably Finite. Shephardson has given such an example. I need some different example. Can anyone supply me another example?
1
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1answer
25 views

Rings and equations

Let $R$ be a commutative (non-zero) ring with identity, what are the solutions of $x^2-1=0$? Obviously, $x=\pm 1$ are solutions and if $R$ is an integral domain there aren't other solutions since $x^2-...
0
votes
2answers
69 views

Suppose that $f:\mathbb{C} \to \mathbb{C}$ is a ring homomorphism. Does $f$ necessarily fix the real axis?

I suspect that only such ring homomorphisms are identity and conjugation, but I cannot see how any homomorphisms from $\mathbb{C}$ to itself fixes the real axis. I have shown that $f$ should fix any ...
0
votes
0answers
47 views

$x^4-x \in Z(R)$ implies commutativity [duplicate]

Let $R$ be a ring and $Z(R)$ the centre of $R$. There exist elementary proofs (that is, proofs not using the structure theory of rings) of the fact that if $x^n-x \in Z(R)$, then $R$ is commutative ...
2
votes
2answers
25 views

need a quick argument for proving ideal non equality

Consider $\mathbb{Z}[x]$. Is there a quick way to argue that $(7) \not\subset (3,x^2+1)$ ? where $(7)$ is ideal generated by $7$ and $(3,x^2+1)$ is ideal generated by $3$ and $x^2+1$ (all ideals are ...
1
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3answers
58 views

Prove that, for each a ∈ R, either $a$ is a zero-divisor or a is a unit.

I know similar question has been asked here before, but the answers were little bit stronger for me to digest :) So, I am asking the same question again. Let $R$ be a finite commutative ring with ...
0
votes
1answer
107 views

ring theory questions… what is a subring?

So I've got coursework to do, and having not been to some (most) lectures, I'm at that time where it's time to learn everything I need to know... Any help is much appreicated, thanks! I've been given ...
0
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1answer
48 views

Show that $Z(XY,XZ,YZ)$ is not irreducible

Show that $Z(XY,XZ,YZ)$ is not irreducible. what I think it is $Z(XY,XZ,YZ)=Z(XY)∩Z(XZ)∩Z(YZ)=(Z(X)∪Z(Y))∩(Z(X)∪Z(Z))∩(Z(Y)∪Z(Z))$ Then I am not sure how to carry on, and what I need to show it is $...
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1answer
337 views

Elements of an annihilator induced by a matrix

An annihilator is defined as $Ann_R(M) = \{r \in R | rm=0 \forall m \in M \}$. However, I read that the minimal polynomial of a matrix $A$ generates $Ann_A(V)$. But, I do not understand; what could be ...
1
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1answer
64 views

Let $F$ be a field and $f: \mathbb{Z} \to F$ be a ring epimorphism.

Here we have $F$ a field and $f: \mathbb{Z} \to F$ a ring epimorphism. We are to prove that $F$ is a finite field with non-zero characteristic. I know that since $f$ is an epimorphism, we have that $...
1
vote
1answer
28 views

Homomorphism from $A[X,Y]$ to $A[X]$ with kernel $(X^i-Y^j)$

Let $A$ be an integral domain, $i,j \in \mathbb N$ such that gcd$(i,j)=1$. How would one define a homomorphism from $A[X,Y]$ to $A[X]$, having the ideal generated by $X^i-Y^j$ as its kernel?
11
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2answers
293 views

Show that $A[X]/(aX+b)$ is an integral domain

Let $A$ be an integral domain, $a$ and $b \in A-\{0\}$, and let $B = A[X]/(aX+b)$. Show that, if $Aa \cap Ab=Aab$, then $B$ is an integral domain. My attempt at proof (following a hint). Denote by $...
1
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1answer
113 views

Image of ring homomorphism $\phi : \mathbb{Z}[t] \to \mathbb{Q}$?

Here is a problem I face practicing the theory of rings: Define $\phi : \mathbb{Z}[t] \to \mathbb{Q}$, a ring homomorphism (it does map $1$ to $1$). I'm trying to show that if $\phi(t)=\frac{u}{v}$...
3
votes
0answers
40 views

are both topologies same

Consider the space $(\mathbb{C^2},T)$ where $T$ is the product topology on $\mathbb{C}$ with $\mathbb{C}$ having Zarisky topology. Now let $T_2$ defines another topology on $\mathbb{C}^2$ with open ...