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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2
votes
1answer
104 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 $p$ ?
2
votes
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
votes
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 ...
2
votes
2answers
73 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
vote
2answers
184 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 ...
1
vote
2answers
29 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.
1
vote
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 ...
4
votes
1answer
177 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) ...
1
vote
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
vote
1answer
63 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 ...
2
votes
1answer
44 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
vote
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
vote
1answer
75 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 ...
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
votes
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
vote
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 ...
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
46 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
vote
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
105 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
votes
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 ...
-1
votes
1answer
313 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
vote
1answer
60 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
27 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
votes
2answers
283 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 ...
1
vote
1answer
112 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 ...
3
votes
0answers
38 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 ...
0
votes
1answer
77 views

If $a^2=a$ in an integral domain, then either $a=0$ or $a=1$

Let $R$ be a commutative ring with zero element $0$ and unity element $1$. Suppose that $A$ is an element of $R$ for which $a^2=a$. Show that if $R$ is an integral domain, then either $a=0$ or ...
2
votes
2answers
91 views

Question on prime ideals of ${\mathbb Z}[x]$

I was thinking this question myself: Consider the topological space $(\text{Spec}(\mathbb{Z}[x]),T )$ where open sets $D_I$ in $T$ are given as indexed by ideals $I$ in $\mathbb{Z}[x]: \; D_I =\{p\in ...
3
votes
0answers
159 views

Is $\mathbb{Z}[x]/(x^3-2)$ a field?

Exercise: Prove or disprove: $\mathbb{Z}[x]/(x^3 − 2)$ is a field. I would say, that it is not a field. We can use the following theorem: Let $R$ be a commutative ring with identity and $M$ ...
0
votes
2answers
49 views

$\phi:M \rightarrow N$ is surjective, then $\hat{\phi}:M/Tor(M) \rightarrow N/Tor(N)$ is surjective

Hello everyone, I already have done that $\phi(Tor(M)) \subset Tor(N)$. I'm stuck on the second part, so this is my attempt. Since $\phi$ is an isomorphism, then since $Tor(M)$ and $Tor(N)$ are ...
2
votes
1answer
110 views

Prime elements in a noncommutative ring

Is there a reasonable definition of prime element in a noncommutative ring? The definition from wikipedia makes the assumption of commutativity and I'd like to know how necessary this condition is. ...
2
votes
1answer
56 views

Set of non-units in non-commutative ring

I consider the question: In a non-commutative ring $R$ with $1$, is the union of maximal left ideals equals the set of non-units? If $x$ is non-unit then it can happen that $yx=1$ for some $y$ ...
0
votes
1answer
46 views

$(0)$ is prime ideal in $M_n(D)$

I am not able to give an elementary proof of the fact that the zero ideal is prime ideal in the matrix ring $M_n(D)$ over a division ring $D$. Any hint? In non-commutative ring $R$, a $2$-sided ideal ...
2
votes
1answer
161 views

Normalization or integral closure of ring over $\mathbb Z_p$

Let $p$ be a prime larger than four. And denote the $p$ adic integers by $\mathbb Z_p$. Consider the ring $A=\mathbb Z_p[x]$ and its field of fractions $K=\mathbb Q_p(x)$. Now let's extend $K$ to a ...
2
votes
2answers
54 views

Describe $\mathbb{Z}[\omega] / (2)$

$$\Bbb Z[ω] = \{\;a + bω: a, b\in\Bbb Z\;\}\;,\;\; ω = e^{2πi / 3} = -\frac12 +\frac{\sqrt3}2i$$ Describe $\;\Bbb Z[ω]/(2)\;$ where $\;(2)\;$ is an ideal. I already described it as a ring, and am now ...
0
votes
1answer
101 views

Ring homomorphisms from $\mathbb{Z}_{12}$ to $\mathbb{Z}_{20}$ [duplicate]

How many ring homomorphisms are there from $\mathbb Z/12\mathbb Z$ to $\mathbb Z/20\mathbb Z$? In all cases, describe the image and kernel of the homomorphism. I am not even sure how to proceed on ...
5
votes
1answer
111 views

Remainder when $x^{1000000}$ is divided by $x^3 + x +1$ in $\mathbb{Z}_2[x]$

I tried the traditional algorithm of long division hoping to find a pattern, but I was not able to. I then tried using the root of $x^3 + x + 1$ $\left(x \sim -0.7\right)$ in the equation: ...
1
vote
0answers
202 views

The Center of a Matrix Ring [duplicate]

Prove that the center of the ring $M_n(R)$ is the set of scalar matrices. I know what a center look like but i feel like i have not enough information to even solve this problem. Anyone that can help ...
2
votes
0answers
74 views

Chain of ideals in nilpotent algebra

Let $R$ be a nilpotent algebra ($R^n = \{0\}$ for some $n \ge 1$) and $A$ be a subalgebra of $R$. I want to show that exist a finite chain of subalgebras {$R_i$ | $i = 0, 1, ..., m $}, $m \ge 1$, ...
0
votes
1answer
41 views

Show that a Set S is a subring of $R \times R$

Question: Prove that$$S=\{ (r,r) | r \in R\}$$ is a subring of $R \times R$. Attempt: Proof: Let $(a,b)$ and $(c,d)$ $\in R$. As $(a,b) \cdot (c,d) = (ac,bd) \in S$. $(a-c, b-d) \in S$. I show ...
0
votes
3answers
44 views

A question in boolean ring theory.

Prove that the only Boolean ring that is an integral domain is $Z/2Z$. I know that the definition of a Boolean ring is $a^2=a$ and that an integral domain is $ab=0$ either $a=0$ or $b=0$. But yet i ...
1
vote
1answer
41 views

If there exists a vertex of $ \Gamma_{2}(R)\setminus J(R) $ which is adjacent to every other vertex then $ R \cong \mathbb{Z}_{2}\times F$

I am reading the research paper Comaximal Graph of Commutative Rings by H.R. Maimani, M. Salimki, A. Sattari, S. Yassemi. In this paper, $ R $ denotes a commutative ring with the identity element. $ ...
0
votes
1answer
58 views

Which subrings $S$ of $\mathbb Z_n$contains a multiplicative identity , that is $\exists e\in S$ such that for every $x \in S , x.e=e.x=x$ ?

I want to find all non-trivial subrings of $\mathbb Z_n$. So let $S$ be a subring (not necessarily containing $[1]$ ). Then $(S,+)$ is a asubgroup of $(\mathbb Z_n,+)$, so $S$ is generated by an ...
1
vote
0answers
31 views

Rings in which every maximal ideal is finitely generated [duplicate]

Suppose that $R$ is a commutative ring with unity in which every maximal ideal is finitely generated. Then is $R$ Noetherian?