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
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1answer
44 views

Let $R$ be a ring and $I$ be a finitely generated nilpotent ideal. If $R/I$ is noetherian (resp. Artinian) then $R$ is so.

Let $R$ be a ring and $I$ be a finitely generated nilpotent ideal. If $R/I$ is noetherian (resp. Artinian) then $R$ is so. In between step is $I^j/I^{j+1}$ is noetherian (artinian) $\forall j$. I ...
1
vote
1answer
27 views

Show pR[x] + (x) is a prime ideal.

I am self studying the notes here. The problem is exercise 2.18 on page 9 (solutions provided there as well). Let R be a ring, p a prime ideal, R[X] the polynomial ring, pR[x] the product ideal ...
0
votes
1answer
36 views

Cartesian Product of Algebras forms an Algebra

Consider the ring $A$ and let $B$ and $C$ be $A$-algebras. I was asked to prove on an old homework assignment that the ring $B \times C$ is an $A$-algebra. From Atiyah and MacDonald, I have the ...
3
votes
0answers
45 views

Show that $R_{1} \times R_{2}$ is not the coproduct of $R_{1}$ and $R_{2}$ in $\mathcal{R}$

Let $\mathcal{R}$ denote the category of rings. Show that $R_{1} \times R_{2} \simeq R_{1} \oplus R_{2}$ is not the coproduct of $R_{1}$ and $R_{2}$ in $\mathcal{R}$. I know if $R_{1} \times R_{2}$ ...
1
vote
1answer
31 views

Is $x^p+p-1$ always irreducible in Q[x] for p prime?

Is $x^p+p-1$ always irreducible in Q[x] for p prime? I have a feeling it is true, however im only able to prove it for p=2,3.How could i generalize it for every p? Thanks
0
votes
1answer
25 views

Confusion regarding finding invariant factors of a matrix.

So I'm having a bit of trouble determining invariant factors of a matrix. Say we have $$ \begin{bmatrix} 2 &0 &0 \\ 0 &9 &0 \\ 0 &0 &6 \end{bmatrix} $$ and I want to find the ...
-1
votes
2answers
47 views

$(a,b)=1$ implies $R/(ab)$ isomorphic to $R/(a) \oplus R/(b)$ [closed]

If $R$ is a PID and $a,b$ belong to $R$ and are relatively prime, then $R/(ab)$ is isomorphic to $R/(a) \oplus R/(b)$ (direct sum). I can't find this problem in Internet. Any idea?
1
vote
1answer
84 views

A non-trivial mapping $\theta : S \to R$

Let $(R,+,\times)$ be a ring with additive identity $0 \in R$. On the set $S = \{(a,b) :\ a,b \in R\}$ the binary operators $\oplus$ and $\otimes$ are defined by: $$(a, b)\oplus(c, d) = (a+c, ...
1
vote
2answers
10 views

Existence of GCD in UFD

I proved that any two elements in PID have GCD and it can be expressed as linear combination of those two elements. I know that even in case of UFD GCD exists but it may not be expressed as linear ...
1
vote
1answer
51 views

how is $m_1 m_2…m_i/m_1 m_2…m_{i+1}$ a vector space over $A/m_{i+1}$? [duplicate]

Please help me with this problem . The problem was used in one of the questions in my examination and I failed to understand. Problem as follows: If $A$ is a commutative ring in which ...
0
votes
1answer
39 views

In an Integral Domain is it true that $\gcd(ac,ab) = a\gcd(c,b)$?

In my algebra class I was given as homework assignment to prove that: Given an integral domain $A$ and $a,b,c,d,e \in A$ then if $d = \gcd(b,c)$ and $e = \gcd(ac, ab)$ then $e = ad$. It is easy ...
0
votes
1answer
33 views

Splitting fields being Galois

For a finite extension $K/F$, $K$ is Galois over $F$ if $\mid Aut(K/F)\mid=[K:F]$. Is the splitting field of any polynomial containing a separable factor Galois?
0
votes
0answers
31 views

Does the tensor product commute with the exterior product?

Let $R,S$ be commutative rings, $S$ an extension of $R$ and $A$ a $R$-module. Is it true that $(\bigwedge^{k} A) \otimes_{R} S = \bigwedge^{k} (A \otimes_{R} S)$? I was trying to use the ...
4
votes
3answers
165 views

Steps to prove or disprove if two rings are isomorphic

So i'm struggling on how to prove if two rings are not isomorphic to one another. My professor told me that if a ring is not isomorphic to another, the best way to prove that this is true is to find a ...
3
votes
1answer
74 views

$p = x^2 + xy + 3y^2$ if and only if $p \equiv 1$, $3$, $4$, $5$, $9$ mod $11$? [duplicate]

For a prime number $p \neq 11$, do we have $p = x^2 + xy + 3y^2$ for some $x$, $y \in \mathbb{Z}$ if and only if $p \equiv 1, 3, 4, 5, 9$ mod $11$? An example where this is true:$$5 = 1^2 + 1 \times ...
0
votes
1answer
26 views

What is the difference between the residue classes / rings of the following polynomial rings mod ideals

I am struggling with fully understanding the difference in structure and elements of the following rings: 1) $R[x]/(x+1)$ 2) $R[x]/(x+2)$ 3) $R[x]/(ax+1)$ I know when we mod by an ideal, the residue ...
2
votes
1answer
27 views

Do there exist “weak ring homomorphisms” that aren't (genuine) ring homomorphisms?

(All my rings have $1$, and all my ring homomorphisms preserve $1$.) Suppose $R$ is a ring. Philosophically, it may be the case that $R$ has too many zero divisors for a particular method "$\mu$" of ...
1
vote
0answers
22 views

Change of scalars and field of fractions

Does someone know a reference in a textbook for the following fact? Let $k \subseteq k'$ be an algebraic field extension and $A$ a $k$-algebra such that the extension of scalars $A \otimes_k k'$ ...
2
votes
4answers
168 views

Are $\mathbb Q\left(\sqrt2\right)$ and $\mathbb Q\left(i\sqrt2\right)$ isomorphic?

I would like to show whether $\mathbb Q\left(\sqrt2\right)$ and $\mathbb Q\left(i\sqrt2\right)$ are isomorphic. I tried noting that $\pm\sqrt2$ and $\pm i\sqrt2$ are the roots of ...
0
votes
2answers
48 views

Properties of integer matrices $A$ such that $A^{p}=I$ for $p$ a prime integer.

Problem Statement: Let $p$ be an integer prime, and let $A$ be an $n\times n$ integer matrix such that $A^{p} = I$ but $A \neq I$. Prove that $n \geq p − 1$. We have been learning factoring of ...
0
votes
0answers
35 views

How many ring homomorphisms would there be from $F_{2}[x]$ to $F_{2}$?

How many ring homomorphisms would there be from $F_{2}[x]$ to $F_{2}$? I think that there is the trivial homomorphism, and a map that sends x to an element of $F_{2}$. So are there 2 ring ...
0
votes
1answer
38 views

Are $\mathbb{Z}[x]/(x^2+2)$ and $\mathbb{Z}[\sqrt{-2}] $ isomorphic as Z modules?

Are $\mathbb{Z}[x]/(x^2+2)$ and $\mathbb{Z}[\sqrt{-2}] $ isomorphic as Z modules? I already know they are isomorphic as rings because $x^2+2$ is the minimal polynomial of $\sqrt{-2}$ However, I'm ...
0
votes
1answer
41 views

Complex numbers of degree less than or equal to 2 over rational numbers

Im trying to figure out which complex numbers have degree $\leq$2 over $\mathbb{Q}$ and then figure out which have degree $\leq$2 over $\mathbb{R}$. For the first question, I know that it is at least ...
2
votes
1answer
27 views

Associated primes of a non-zero module always exist if corresponding ring is Noetherian

Let $R$ be a commutative Noetherian ring. Let $M$ be a non-zero $R$-module (not necessary finitely generated). Prove that set of associated primes of the module is not empty, ...
4
votes
1answer
94 views

Show that a radical ideal has no embedded prime ideals. [closed]

Let $A$ be a commutative ring and $I$ a decomposable ideal. Let $I=\bigcap_{k=1}^{n} I_k$ be a minimal primary decomposition. Show that if $I=\sqrt{I}$ then $I$ has no embedded prime ideals. (I ...
22
votes
7answers
2k views

How to prove that a complex number is not a root of unity?

$\frac35+i\frac45$ is not a root of unity though its absolute value is $1$. Suppose I don't have a calculator to calculate out its argument then how do I prove it? Is there any approach from ...
0
votes
0answers
47 views

When do the characters of an integral domain form an algebraically independent subset?

Work over a fixed commutative ring $R$. I'll just make sure we're all on the same page by quickly stating the relevant definitions: Definition 0. Given an $R$-algebra $X$ together with a subset ...
2
votes
0answers
21 views

Transcendental extension not isomorphic to its closure

Suppose I'm given a field extension $K/F$ with $\alpha\in K$ transcendental over $F$, the claim is that $F(x)\cong F(\alpha)$. It's a statement without proof in our class notes, and the remarks ...
1
vote
2answers
18 views

The order of all the elements of a finite additive group of a ring divides characteristic

Let $(R,+,\cdot)$ be any ring, $|R|<\infty$. We know that $\forall a\in (R,+),\, ord(a)<\infty$ . 1) How can we prove that: $$\forall r\in(R,+,\cdot),\, ord(r)|Char(R) \ \ \ \ (\dagger)$$ ...
1
vote
1answer
140 views

How to show that $(S, \oplus, \otimes)$ is a ring? [closed]

Let $(R,+,\times)$ be a ring with additive identity $0 \in R$. On the set $S = \left.\left\{ (a,b) \,\right|\, a,b \in R \right\}$ the binary operators $\oplus$ and $\otimes$ are defined by: $(a, b) ...
-3
votes
1answer
49 views

Let $A$ a ring such that $0=m_1m_2…m_k$ where $m_i$ are maximal ideals of $A$. [closed]

Let $A$ a ring such that $m_1m_2...m_k=0,$ where $m_i$ are maximal ideals of $A$. Then $A$ is Noetherian if and only if $A$ is Artinian.
0
votes
2answers
21 views

If N and M/N are free A modules on the ring A with 1, then M is also a free A module

I would like to proove the following : If N and M/N are free A modules on the ring A with 1, then M is also a free A module. My main issue is that we dont know if N and M/N have a finite rank. ...
6
votes
2answers
166 views

Which are integral domains? Fields?

Which of the following rings are integral domains? Which ones are fields? (a) $\mathbb{Z}[x]/(x^2 + 2x +3)$ (b) $\mathbb{F}_5[x]/(x^2+x+1)$ (c) $\mathbb{R}[x]/(x^4+2x^3 +x^2 +5x+2)$ For (a), ...
0
votes
0answers
22 views

is it true that closures preserve isomorphisms [duplicate]

Suppose I have two isomorphic integral domains $A$ and $B$. Are the fields of fractions of these two rings isomorphic as well?
1
vote
1answer
87 views

Prove that $\mathbb Q[\sqrt[3]2]$ is a field

We define the set: $$\mathbb{Q}[\sqrt[3]2]=\{a_{0}+a_{1}\sqrt[3]{2}+a_{2}\sqrt[3]{2^{2}}:a_{0}, a_1,a_2\in\mathbb{Q}\}$$ It's easy to prove all the properties of fields, except for the unit ...
0
votes
2answers
46 views

The polynomial of minimal degree with root $\alpha$ is unique.

So I am working on the following proof: Problem Statement: Let $\alpha$ be a complex number. Prove that the kernel of the substitution map $\mathbb{Z}[x] \rightarrow \mathbb{C}$ that sends $x ...
0
votes
1answer
36 views

Well-Defined Maps on Equivalence Classes of Rings

Let $R$ be a unitary commutative ring such that $1 \neq 0$ and $S \subseteq R$ be closed under multiplication (i.e. $\forall x,y \epsilon S, xy \epsilon S$) and contain 1. We define the relation $E$ ...
2
votes
1answer
40 views

Structure of $\mathbb{Q}S_3$

I have an exercise to show that $\mathbb{Q}S_3 \cong \mathbb{Q} \oplus\ \mathbb{Q}\ \oplus\ M_2(\mathbb{Q}) $ , where $M_2(\mathbb{Q})$ is ring of $2$ by $2$ rational matrices and $\mathbb{Q}S_3$ is ...
1
vote
2answers
45 views

Ring isomorphism, very basic question(regarding first isomorphism theorem)

Feeling unsure, want to check if I am thinking correctly. Exercise sounds: Let n>1, square-free integer. Show that $\mathbb{Z}[\sqrt{n}]/\langle \sqrt{n}\rangle\simeq\mathbb{Z}_n$ My take: Lets make ...
0
votes
1answer
18 views

Hilbert's Nullstellensatz - question about generalization

It is well-known that if $k$ is algebraically closed field that has infinite transcendence degree over the prime field $\mathbb{Q}$ or $\mathbb{F}_p$ then the maximal ideals of $k[x_1,...,x_n]$ are of ...
0
votes
0answers
30 views

Conditions for a ring to be a direct product of local rings

I recently came across a property of commutative rings which I could prove only for rings that are (isomorphic to) a direct product of (possibly infinitely many) local rings. It might be that my ...
1
vote
1answer
29 views

Jacobson radical of polynomial quotient ring.

Let $F$ be a field, and $A=F[x]/(x(x-1)^2)$. 1. Find the ideals of $A$. Which of them are simple or maximal? 2. Find the Jacobson radical, $J(A)$, of $A$. 3. Find two composition series for $A$, as ...
0
votes
0answers
16 views

About endomorphism ring

Is there a natural way to embed $ R \otimes_Z K $ to the endomorphism ring of $ R $ when $ R $ is considered as a vector space of its center $ Z $? Here, $ R $ is a division ring and $ K $ is a ...
0
votes
0answers
24 views

Etale over a product implies etale over factor?

I will say an $R$-algebra, $A$, is finite étale over a commutative (with unity) ring $R$ if there exists some (faithfully-flat) $R$-algebra, $W$, such that $A\otimes_{R} W \cong W^{n}$ as a ...
0
votes
0answers
18 views

Are all integral domains in which all irreducible elements are prime G.C.D domains?

I know that in G.C.D domains all irreducible elements are prime. Does the converse of this statement hold? If not, is there a weaker condition than being a G.C.D. domain that is both sufficient and ...
1
vote
4answers
41 views

How to show $\text{Im}~ \theta=\Bbb{Q} [\sqrt 2]$ for a homomorphism?

How to show $\text{Im} ~\theta=\Bbb{Q} [\sqrt 2]$ for the homomorphism defined as $\theta:\mathbb{Q}[X] \rightarrow \mathbb{R}$ given by $\theta(f(X))=f(\sqrt2)$. I can show $\Bbb{Q} [\sqrt 2] ...
4
votes
1answer
41 views

What are the kernels of ring homomorphisms that contain the polynomial $X^2+1$?

Let $\mathbb{R}[X]$ be the ring of polynomials over the field of real numbers. Write down two ring homomorphisms $\mathbb{R}[X]\rightarrow\mathbb{C}$ whose kernels contain the polynomial ...
4
votes
1answer
57 views

Prove that the following are isomorphic as groups but not as rings

$\mathbb{Z}$ and $\mathbb{2Z}$ My solution: To prove they are isomorphic as groups, I take the mapping $f: \mathbb{Z} \rightarrow \mathbb{2Z}$ defined by $f(x)=2x$. I prove it's a homomorphism and ...
0
votes
1answer
24 views

$a$ is algebraic over $k(b)$ where $b=g(a)$ for some non constant polynomial $g$ [closed]

Let $k \subset K$ be a field extension and $a \in K$. Show that if $g \in k[x]$ is any nonconstant polynomial and $b = g(a)$, then $a$ is algebraic over $k(b)$.
2
votes
2answers
62 views

An example of ideal $I$ such that $I^{ec}\neq I$

Let $A$ be a commutative ring, $S \subseteq A$ a multiplicative system and $i_S : A \rightarrow S^{-1}A$ the canonical morphism. Can you give me an example of ideal $I \unlhd A$ such that $I^{ec}\neq ...