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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Kernel of ring homomorphism is an ideal

I am asked to show that if f is a ring homomorphism from R to R' then kernel of f is an ideal of R. According to definition of ideal : A non empty subset of R is an ideal for any two elements of ...
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1answer
46 views

Tensor product of coordinate rings corresponds to pullback

Here in Milne's notes on algebraic geometry, he proves that if $k$ is an algebraically closed field, and $A$ and $B$ are reduced finitely generated $k$ algebras, then $A \otimes_k B$ is reduced. (This ...
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2answers
32 views

In the ring of Gaussian integers $\mathbb{Z}[i]$, show that $2\in J$ where $J$ is the ideal $\langle i + 1\rangle$

In the ring of Gaussian integers $\mathbb{Z}[i]$, show that $2 \in J$ where $J$ is the ideal $\langle i + 1\rangle$. My thought was that $(1+i) \in J$ and $(1-i) \in \mathbb{Z}[i]$. Then if $\langle ...
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1answer
36 views

Prove or disprove that the ring $\mathbb{Z}[x]/(x^2-1)$ is an integral domain

Prove or disprove that the ring $\mathbb{Z}[x]/(x^2-1)$ is an integral domain. It is easy to see that $\mathbb{Z}[x]/(f)=\{P+(f): \deg(P)<\deg(f)\}$. If $\deg(P) \geq \deg(f)$, there exists ...
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2answers
37 views

What are the principal ideals of $\mathbb Z$

What are the principal ideals of $\mathbb Z$ I thought the answer would be $\mathbb Z$ and $\{0\}$. However, the answer says: $m\Bbb Z \subseteq\Bbb Z $ for $m > 0$ Can somebody explain why? ...
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0answers
16 views

Maximal $\mathbb Z_p$-orders in $\mathbb Q_p[G]$

Let $G$ be a finite group and $S$ a finite set of prime numbers. I know that every separable $\mathbb Q$-algebra $A$ contains a maximal $\mathbb Z$-order but I wonder if the following is true. ...
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1answer
18 views

Probability of nonsingular matrix over finite ring

Let $p$ be a prime and $p|q$. Let $m $ and $n$ be integer$(m>n)$. Consider a ring $\mathbb Z_p$ and $\mathbb Z_q$.(modulus $p$, modulus $q$ ,respectively) Let R be a ring and $A \in Mat_{m \times ...
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0answers
44 views

How injective $\overline{f_p}$ maps $\mathfrak{m}M_{p}/\mathfrak{m}^2M_{p}$ to $\mathfrak{m}L_{p}/\mathfrak{m}^2L_p$?

If $(R,\mathfrak{m},k)$ is a local ring, $A$ a finite $R$-module. Let $L_{\bullet}:\cdots\rightarrow L_1\xrightarrow{d_1} L_0\xrightarrow{d_0} A\rightarrow 0$ be a minimal free resolution. $\bar{d_i}$ ...
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2answers
47 views

Determining if a subgroup is an ideal

I need some clarification on this problem. These are my answers and I was wondering if they are correct or not. Any help is appreciated. Determine whether the indicated set A is an ideal in the ...
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1answer
19 views

Let $F=\mathbb{Z}/2\mathbb{Z}$. Show that $F[x]/(x^2+1)$ is a ring of four elements

Let $F=\mathbb{Z}/2\mathbb{Z}$. Show that $F[x]/(x^2+1)$ is a ring of four elements. I thought I could use the first isomorphism theorem, but I get stuck. Could anyone help me at this point?
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1answer
29 views

Showing that two rings are homomorphic to one another.

For $a, b ∈ \Bbb Z$, let $B(a, b) ∈ M(2, \Bbb Z)$ be defined by $B(a, b) = \begin{bmatrix} a & 3b \\ b & a \end{bmatrix}$. Let $S = \{B(a, b) | a, b ∈ \Bbb Z\} ⊆ M(2, Z)$. Show that $S ...
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2answers
42 views

Isomorphisms preserving integral domains and fields

Let $R \simeq S$ be isomorphic commutative rings with unity. Prove the following: a). If $R$ is an integral domain then $S$ is an integral domain. For this, I said that If I let $f: R \to S$ be an ...
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1answer
18 views

If $A$ is a commutative, unitar ring and $I$ an ideal of $A$ such that $I$ and $A/I$ are Noetherian rings, then $A$ is Noetherian?

If $A$ is a commutative, unitar ring and $I$ an ideal of $A$ such that $I$ and $A/I$ are Noetherian rings, then $A$ is Noetherian ? I know just that if $A$ is Noetherian then $A/I$ is Noetherian.
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1answer
30 views

Atiyah-Macdonald, Exercise 4.6 [duplicate]

Let $X$ be an infinite compact Hausdorff space and let $C(X)$ be the ring of real-valued continuous functions on $X$. Does $(0)$ have a primary decomposition in this ring? I feel like the answer ...
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0answers
7 views

Showing a ring is a subring, has unity, and is a division ring [duplicate]

The quaternion group $Q_8$ is the group which consists of the following eight matrices in $M(2, \Bbb C):Q_8 = \{I, A, A2, A3, B, BA, BA^2, BA^3\}$. $A = \begin{bmatrix}0 & 1\\-1 & ...
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1answer
28 views

gaussian integers quotient ring

Let $R$ denote the factor ring $\mathbb{Z}[i]/(1+3i)$. Show that $i-3 \in (1+3i)$ and that $i+(1+3i) = 3 + (1+ 3i)$ in R. I am unsure how to find the elements of this factor ring? I know how to find ...
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2answers
121 views

How can we find the prime ideals?

I have found that the maximal ideals of the ring $\mathbb{Z}_{12}$ are $(2)$ and $(3)$. Is this correct? How can we find the prime ideals of $\mathbb{Z}_{12}$ ?
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1answer
44 views

Localization Preserves Euclidean Domains

I'm wanting to prove that given a ring $A$ (by "ring" I mean a commutative ring with identity) and a multiplicative subset $S \subset A$: if $A$ is an Euclidean Domain, and $0 \notin S$ then ...
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1answer
35 views

Product of two principal ideals in $\Bbb Z[x]$

I'm looking for an easy argument for the following question: True or false, and why: The product of two principal ideals in $\Bbb Z[x]$ is a principal ideal. I know that $\Bbb Z[x]$ is not a ...
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0answers
21 views

Commutative rings and PI algebras

Any commutative ring $R$ with unity is a PI ring (polynomial identity ring). When could one take $R$ as a PI algebra? Essentially, what is the relation between an arbitrary commutative ring and a ...
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2answers
26 views

Dimension of a basis for a module using the quotient map

I'm going through a proof that the size of a basis for a module is uniquely determined and can't see why the following line from my lecture notes follows: "We now claim that if $X$ is a basis for ...
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1answer
40 views

How to compute unit groups in quotient rings? [closed]

How do I compute the unit group of a quotient ring $\mathbb{R}[x]/(f(x)\mathbb{R}[x])$, for example $f(x)=x^2+2x+1$?
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1answer
35 views

$A\subset B$ if $A\cdot R[X] \subset B \cdot R[X]$? [closed]

Can we conclude $A\subset B$ if $A\cdot R[X] \subset B \cdot R[X]$ for ideals $A,B$ in $R$, where R is a commutative ring with unity and $A \cdot R[X]$ the ideal generated by the products $af$, ...
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0answers
14 views

Checking Euclidean Domain [duplicate]

Prove that Z[i] is an Integral Domain hence prove that Z[i] is an Euclidean Domain.
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1answer
31 views

$N$ is $(0)$ if $p(x)$ is not divisible by the square of any polynomial

I am trying to show that in the ring $R=\frac{F[x]}{\langle p(x)\rangle}$, the set of nilpotent elements, $N$ is $(0)$ iff $p(x)$ is not divisible by the square of some polynomial $q(x) \in F[x]$. I ...
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2answers
46 views

Proving $f(x)$ is not a square in $k[x]$

Let the field $k$ be algebraically closed, let $f(X) \in k[X]$ be a separable polynomial of degree at least $2$, let $$ B = \frac{k[Y,X]}{(Y^2 - f(X))} $$ and write $y,x$ for the images in $B$ of $Y$ ...
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1answer
22 views

Find a vector $v$ such that $V=\mathbb{C}[T]\cdot v.$

Let $V=\mathbb{C}^2$ and let $\alpha:V\to V$ be the $\mathbb{C}$-linear map given by \begin{equation*} \begin{pmatrix} a\\ b \end{pmatrix} \longmapsto \begin{pmatrix} 2 & 3\\ -3 & 8 ...
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0answers
16 views

When does the equality $\mathrm{ht}\:\mathfrak{p}+\mathrm{coht}\:\mathfrak{p}=\dim R$ happen? [duplicate]

In the context of Krull dimension, given any commutative ring $R$ and $\mathfrak{p}\subset R$ a prime ideal, we have (almost by definition) $$ \mathrm{ht}\:\mathfrak{p}+\mathrm{coht}\:\mathfrak{p} ...
2
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1answer
21 views

Let $A$ be a principal ideal domain, and $a,b,d$ elements of $A$. Prove that $d$ is a gcd of $a$ and $b$ if and only if $aA+bA=dA$.

I can prove that $aA+bA=dA$ implies that $d$ is a gcd of $a$ and $b$. I can also prove that $d$ being a gcd of $a$ and $b$ implies that $aA+bA\subset dA$, since $a+b$ is a multiple of $d$. What im ...
3
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1answer
69 views

Exercise on radical ideal and formal derivatives

I need some help for solving the following exercise, because at the moment I'm a little bit lost and don't know where to start. Given a field $k$ with $\mathrm{char}(k)=0$ and a polynomial $f\in ...
3
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2answers
86 views

Show that some ideals are prime

I want to show that the ideals $(x,y)$ and $(2,x,y)$ of $\mathbb{Z}[x,y]$ are prime ideals. Could you give some hints how we could do that? Do we have to show that $\mathbb{Z}[x,y]/(x,y)$ ...
4
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1answer
136 views

$M(A)=\left\{g\in G:g^n=\prod_{i=1}^ma_i^{n_i},a_i\in A,n_i\in\mathbb N\cup\{0\},n\in\mathbb N\right\}?$

Let $G$ be a group and $\emptyset\neq A\subseteq G$. Let $$M(A)=\bigcap_{A\subseteq C}C$$ where $C$ is any subset of $G$ such that $c^k\in C$ for all $c\in C$ and $k\in\mathbb N\cup\{0\}$ and if ...
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2answers
27 views

Confirming some notation regarding Ring of Polynomials

Hi I just want to clear up some confusion regarding some notation. If $R$ is a ring and $\mathfrak p$ is a prime ideal in $R$, does $(R/\mathfrak p)[x] = R[x]/\mathfrak p[x]$? (or perhaps they ...
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2answers
58 views

$\operatorname{char}R=0 \implies\mathbb{Q} \hookrightarrow R$

Let $R$ be any field, then: $$\operatorname{char}R=0 \implies \mathbb{Q} \hookrightarrow R$$ Proof: We know that $\mathbb{Q} = Q(\mathbb{Z})=\{[(x,y)]\subseteq\mathbb{Z}\times \mathbb{Z^*}:(x,y) ...
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2answers
34 views

Let $D$ be a principal ideal domain. Show that every proper ideal of $D$ is contained in a maximal ideal of $D$.

I know that a PID must satisfy the Ascending chain condition. So Im guessing its going to involve that in the argument some way but Im not sure how to prove it.
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0answers
26 views

How does $\mathbb{Z}_2[x]/f\mathbb{Z}_2[x]$ look like?

I have quite easy question. Is working in $\mathbb{Z}_2[x]/ f\mathbb{Z}_2[x]$ the same as working in $\mathbb{Z}_2[x]$ modulo polynomial $f$ ($\in \mathbb{Z}_2[x]$)? I have quite problems with the ...
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2answers
31 views

Reduced ring are SI?

A ring $R$ is called an SI-ring if for any $a\in R$ the right annihilator of $a$ is an ideal of $R$. It is equivalent to the following statement: "if $ab=0$ for $a,b\in R$ then $aRb=0$". Is it true ...
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3answers
82 views

Motivation for rings of fractions?

I'm learning about rings of fractions and localization. I like the material a lot and feel engaged with it, but I do lack a broader perspective on things. For example, I'm aware of things such as ...
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1answer
44 views

$R$ is isomorphic to a direct product of matrix rings over division rings

Suppose as rings, $R$ is isomorphic to a direct product of matrix rings over division rings, that is $R=R_1 \times ... \times R_n$ where $R_i$ is a two-sided ideal of $R$ and $R_i$ is isomorphic to ...
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23 views

Constructing Witt polynomials

I am reading about Witt vectors, and I keep seeing the following set of congruences often: For example, in these notes here, on page 3 we see the following congruences: \begin{align} ...
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1answer
31 views

Determine if the subset S is a subring of C(R)

determine if the subset $S = \{ f \in \mathcal{C}(\mathbb{R}): \int_0^1f(x)\;dx= 0 \}$ is a subring of $\mathcal{C}(\mathbb{R})$. I know to prove this, I must show that: $S$ is nonempty, $S$ is ...
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37 views

Determine if $g$ is a ring homomorphism

Define $g : ℤ_3 \times ℤ_3$ by $g (a,b) = a +_3 b$ with the usual operations of $+_3$ and $\times_3$, determine if $g$ is a ring homomorphism. I know that I need to show that $g ( a + b) = g (a) + ...
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3answers
77 views

Why is $\mathbb{Z}[\sqrt{-4}] = \mathbb Z[2i]$ not a UFD?

Why is $\mathbb{Z}[\sqrt{-4}] = \mathbb Z[2i]$ not a UFD? $\mathbb Z[2i] = {a+bi}$ where $b$ is even. How do I show this since $4$ is not square-free?
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1answer
41 views

$\mathbb{Z}_p \hookrightarrow R$ is it necessary $R$ commutative?

Let $R$ be an integral domain with $\operatorname{Char}(R)=p$, with $p$ prime. Then: $$\mathbb{Z}_p \hookrightarrow R$$ The proof is not difficult. My questions are: 1) Is it necessary to have an ...
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1answer
17 views

Do the Categories of Rings With a Particular Characteristic Have Terminal Objects?

For a given characteristic n, is there a unital ring T of characteristic n such that for any other unital ring R of characteristic n, there is a unique ring homomorphism from R to T? It's easy to see ...
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1answer
68 views

Units of $\mathbb Z[X]/(X^n+1)$?

What are the units of the cyclotomic ring $\mathbb Z[X]/(X^n+1)$, with $n$ being a power of $2$? I am starting to think that the set $\{\pm X^k,k=0,\dots,n-1\}$ contains all units, is that so ?
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1answer
51 views

Why is $\mathbb{Z}[\sqrt{-5}]$ an integral domain? [duplicate]

I could use some help with this. I know that $\mathbb{Z}[\sqrt{-5}]=\{a+b\sqrt{-5\} }|a,b\in\mathbb{Z}\}$. I then put $$0=(a+b\sqrt{-5})(c+d\sqrt{-5})=ac-5bd+(ad+bc)\sqrt{-5}$$ which leaves me with ...
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0answers
18 views

Double cosets (Neukirch's Algebraic Number Theory)

This is a question from Neukirch's Algebraic Number Theory, Ch.1 $\S$9. Let $A$ be a Dedekind domain with quotient field $K$, $L$ a finite separable field extension of $K$ and $B$ the integral ...
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2answers
52 views

Is every Noetherian Ring an integral domain? [closed]

And if this is so, could you please provide an example of an integral domain which is not a noetherian ring, and also a noetherian ring which is no unique factorization domain? This would be really ...
2
votes
1answer
17 views

Matrix closed under subtraction

I am trying to show that the set $$\left\{\begin{pmatrix}a&b\\0&0\end{pmatrix}\,\middle|\; a,b\in\mathbb Z\right\}$$ is a subring of $M_2(\mathbb Z)$ (the $2\times2$ matrix ring over the ...