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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3
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1answer
54 views

Is every well ordered commutative nontrivial ring with identity an well ordered integral domain?

$\mathbb Z$ is up to ring isomorphism the only well ordered domain, that is, $\mathbb Z$ is a integral domain and every nonempty subset of $\{n \in \mathbb Z: n\geq 0\}$ has a least element. But what ...
0
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2answers
31 views

How to determine if this is a principal ideal domain?

Consider $\mathbb{Z[\sqrt{-5}]}=a+b\sqrt{-5}$ where $a,b \in \mathbb{Z}$. My understanding is that an integral domain is a PID if every ideal in the ring is principal. For the above example, this I ...
1
vote
2answers
57 views

Show that $a(-1) = (-1)a = -a $.

In a ring $R$ with identity 1, show that $$a(-1) = (-1)a = -a \qquad\forall\, a \in R$$ I have started with $a + (-a) = 0$ but cant proceed from here.
0
votes
1answer
30 views

Q has no maximal subgroups.

Theorem: If $R$ is a ring with 1 and $I$ is a ideal in $R$ such that $I \neq R$, then there is maximal ideal $M$ of the same kind as $I$ such that $I\subseteq M$. Note:- IF $R$ has no unity it is not ...
1
vote
1answer
23 views

I've proved everything about the ideal correspondence easily except $\pi ^{-1} \pi (\frak{a}) = \frak{a}$

The correspondence theorem to which I refer is the bijection between ideals of a commutative ring with $1$, $A$, and ideals of $A/\frak{b}$. I can prove easily most parts that imply the bijection ...
0
votes
1answer
36 views

Sum of Two Squares in Ring Theory

Show that a prime $p$ in $\mathbb{Z}$ is a sum of two squares iff -1 is a square in $\mathbb {Z}_{p}$. This example belong to my ring theory book didnt have ideal. i read in number theory that If ...
2
votes
0answers
38 views

Why is every non-zero element not a unit of this ring?

Consider the ring $\mathbb{Z}[\sqrt2]=a+b\sqrt2$ where $a,b\in\mathbb{Z}$. Now, if I am understanding the definition of units correctly, they are all the elements within the ring that have ...
0
votes
0answers
18 views

multilinear identity of degree 2

Let F be any field. A multilinear identity in $m$ indeterminates is an identity which has the form: $$f(x_1,x_2,\dots,x_m)=\sum_{\sigma\in S_n}a_{\sigma}x_{\sigma(1)}x_{\sigma(2)}\cdots ...
0
votes
1answer
49 views

Find all ring homomorphisms from $\Bbb Q$ to $\Bbb R$ [duplicate]

My question is find all homomorphism $ f: \Bbb Q \to \Bbb R$. I think I should use ring isomorphism theorem to do this problem, but I just don't know how to do this.
1
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1answer
41 views

In Ring Theory, does a 'power' of a morphism represent composition?

Say there is a ring homomorphism, denoted by $\theta$. If the notes use the expression $\theta^2$, then are they referring to the composition of the $\theta$ homomorphism with itself?
-1
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0answers
27 views

Prime Factorization of 6

What would be the prime factorization of 6 in $Q[√−1]$? Can I generalize this to other numbers as well or no? Can someone please help me here?
1
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1answer
25 views

Quadratic Prime integer norm is not prime

What would be an example of a quadratic integer in $Q[√−1]$ which is prime, but whose norm is not prime?
3
votes
1answer
71 views

What does the notation $\mathbf{R}^\mathbf{R}$ mean?

I was reading the Princeton Review of GRE math subject test (4th edition), and one question was (page. 251) Example 6.24 Is the ring $\mathbf{R}^\mathbf{R}$ an integral domain? ...
1
vote
4answers
64 views

Ring homomorphism from $M_3(\mathbb{R})$ into $\mathbb{R}$

I was working on this problem Find all ring homomorphisms from $M_3(\mathbb{R})$ into $\mathbb{R}$. My attempt:- I found that if we have any ring homomorphism $\phi$, then $\ker(\phi)$ should ...
0
votes
1answer
44 views

Let I be an unmixed radical ideal of R. then (I:x) is unmixed

Let $R$ be commutative ring with $1$. One says that an ideal $I$ is unmixed if $I$ has no embedded prime divisors (in other words, if the associated prime ideals of $R/I$ are the minimal prime ideals ...
0
votes
2answers
43 views

is 0 in the following Ideal?

Given $R=\mathbb R[x]$ and $I=(2x^3-3x^2+2x-3)+(2x^2-x-3)$ Is an Ideal of R? I don't understand what the quantity I is... Am I supposed to sum them together giving $2x^3-x^2+x-6$ Now here's the ...
4
votes
1answer
30 views

Quadratic number field which is Euclidean but not norm Euclidean

I am looking for an example of a quadratic field $\mathbb Q[\sqrt d]$ , with $d \equiv 2 $ or $3(\mod 4)$ , whose ring of integers is Euclidean but not norm (http://en.wikipedia.org/wiki/Field_norm ) ...
0
votes
1answer
32 views

Finite field extension over the rationals need only one generator?

Some book stated (without proof) that in every finite dimensional field over the rationals of dimension $n$, there is an element of degree $n$ (i.e. any field $Q[\alpha_1, ..., \alpha_n]$ is of the ...
2
votes
2answers
36 views

Nilpotent elements in the quotient ring of a polynomial ring

If $F$ is a field and $p(x) \in F[x]$, prove that the ring $R=F[x]/(p(x))$ has no nonzero nilpotent elements iff $p(x)$ is not divisible by the square of any polynomial. (==>) $R$ has no ...
1
vote
1answer
37 views

About ring $( \frac{F[t]}{t^2})$

I was trying to study some properties of the group $GL_2 ( \frac{F[t]}{t^2})$ where $F$ is a field with $p$ elements. $p$ is a prime. I found that $ \frac{F[t]}{t^2}$ is a discrete Valuation ring. ...
1
vote
1answer
40 views

show that $g(x)= x^n f(\frac 1x) \in \Bbb F[x]. $ where $\Bbb F $ is a field

Let $\Bbb F $ be a field and $f(x)=\sum_0^n a_i x^i \in \Bbb F[x]$. Show that $g(x)= x^n f(\frac 1x) \in \Bbb F[x]$ Show that if $r \neq 0$ is a root of $f(x)$ then $r^{-1}$ is a root of $g(x)$ Find ...
1
vote
1answer
14 views

Isomorphism of tensor product involving a principal ideal

This question arose when dealing with a long exact sequence of Tor. Let $R$ be a (not necessarily commutative) ring, $g$ a central element of $R$ and $M$ a right $R$-module. We have an exact sequence ...
3
votes
2answers
28 views

Is it true that an integral domain $R$ is a UFD if and only if intersection of any two principal ideals of $R$ is principal ?

Is it true that an integral domain $R$ is a UFD if and only if intersection of any two principal ideals of $R$ is principal ?
0
votes
1answer
55 views

If every maximal ideal is finitely generated is the ring Noetherian? [duplicate]

$R$ is a commutative ring with $1$. Suppose every maximal ideal is finitely generated. Is this ring Noetherian? Equivalently, is every prime ideal finitely generated?
1
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1answer
53 views

Annihilating Ideal of a Ring

I am stuck on how to show this. A starting hint would be helpful, and an answer (hidden) would be much appreciated. I tried supposing that there was another element in the annihilating ideal, however, ...
2
votes
1answer
77 views

Functorial construction with two integral domains

Motivated by this question: Let $\mathsf{Int}$ be the category of integral domains with ring homomorphisms (perhaps only injective ring homomorphisms, if you need this). Is there a functor ...
1
vote
1answer
31 views

A small doubt in group rings.

Let RG be a group ring then if $r \in R$ and $g \in G $ then why $rg=gr$ in RG? What does the author means here. Why does these embeddings implies $rg=gr$
3
votes
1answer
50 views

Is this an Integral Domain?

Let X be a non-empty set, and set $P(X)$ the set of all subsets of X with addition and multiplication: $A+B = (A \cup B) \setminus (A \cap B) $ and $A \cdot B = A \cap B $ I am just ...
0
votes
2answers
27 views

Consider the ring homomorphism $ϕ : \mathbb{R}[x] → \mathbb{R}[\sqrt{−3}]$ defined by $ϕ(x) = \sqrt{−3}$.

Consider the ring homomorphism $ϕ : \mathbb{R}[x] → \mathbb{R}[\sqrt{−3}]$ defined by $ϕ(x) = \sqrt{−3}$. i) Show that $ϕ$ is surjective. It seems obvious, so not sure how to show it ii) Find $\ker ...
0
votes
1answer
40 views

Let $R$ be a PID. Prove that $\exists c \in R$ such that $c\mid a, c\mid b$ and $c = ax + by$.

Let $R$ be a PID and $a,b \in R$. Prove that $\exists c \in R$ such that $c\mid a, c\mid b$ and $c = ax + by$ for some $x,y \in R$.
0
votes
1answer
54 views

The Jacobson radical under maps [closed]

let $f:R\to S$ be a surjective morphism of rings. Is $f(J(R))$ a subset of $J(S)$? Note that $J(R)$ denotes the Jacobson radical of $R$.
0
votes
2answers
47 views

True or false question about polynomial ring

Let, $\mathbb{R}[x]$ be a polynomial ring and let $J = (x)$. True/false: $J$ consists of all the polynomials of $\mathbb{R}[x]$ whose constant terms are $0$. I know $J=(x)$ is a maximal ideal of ...
1
vote
1answer
88 views

describe the ring $R=M_2(\Bbb F)$, where $\Bbb F$ is a field

Let the ring $R=M_2(\Bbb F)$, where $\Bbb F$ is a field. What is the description $R^*$. Find order $(M_2(\Bbb Z_3))^*$.
0
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2answers
30 views

Is the kernel of a ring homomorphism a subring?

The following link proves that if $f:R \to R'$ is a ring homormorphism, then $\ker(f)$ is a subring of $R:$ https://proofwiki.org/wiki/Kernel_of_Ring_Homomorphism_is_Subring But an alternative ...
0
votes
1answer
23 views

Prove the field of fractions of $F[[x]]$ is the ring $F((x))$ of formal Laurent series.

Prove the field of fractions of $F[[x]]$ is the ring $F((x))$ of formal Laurent series. $F[[x]]$ is contained in $F((x))$. So there's at least a ring homomorphism that is injective. Can also see ...
2
votes
1answer
41 views

Determine the units of the ring $A= \mathbb Z[X]/(X^3)$ and the structure of the group $A^*$ [duplicate]

Determine the units of the ring $A= \mathbb Z[X]/(X^3)$ and the structure of the group $A^*$ I've only managed to show that the free coefficient of any unit in $A$ is a unit in $\mathbb Z$.
1
vote
1answer
51 views

Maximal ideal of $\mathbb{R}[x]$

Let $\mathbb{R}[x]$ be a ring and let $J = (x)$. Prove that $J$ is a maximal ideal of $\mathbb{R}[x]$
2
votes
1answer
46 views

Subring of a commutative Noetherian ring

We know that it's possible subring of the commutative Noetherian ring become not Noetherian (for example: Subring of a Noetherian ring need not be Noetherian?). But if $S$ be a subring of ...
0
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0answers
36 views

Simple algebra that is not a simple ring

maybe this question is trivial, however I'm not acquainted with non-commutative stuff. In http://www.encyclopediaofmath.org/index.php/Simple_algebra, it's written that a simple algebra may not be a ...
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2answers
44 views

$x^3+ (5m+1)x+ 5n+1$ is irreducible over $\Bbb Z$

How to prove that the polynomial: $x^3+ (5m+1)x+ 5n+1$ is irreducible over the set of integers for any integers $m$ and $n$? I was trying to put $x= y+p$ for some integer $p$ so that I could apply ...
0
votes
1answer
31 views

ordering two rational functions

I have read that $$\frac{x^2 +3}{2x+1}$$ is less than $$\frac{2x-1}{2x+1}$$ in an ordered field,in $\mathbb{Q}((x))$, but how is that result computed? How do we compare two rational fractions like ...
4
votes
1answer
21 views

Module isomorphism from $R$ to $R \oplus R$ for a certain ring $R$

My textbook says: Let $R$ denote the set of infinite–by–infinite, row– and column–finite matrices with complex entries. Show that $R \cong R \oplus R$ as $R$–modules. So for $A, B \in R$, I tried ...
7
votes
2answers
163 views

What's the theoretical basis for integration using partial fractions?

Exercises involving integration using partial fractions depend on expressing a rational function $\frac{P(x)}{Q(x)}$ (where the degree of $P$ is less than the degree of $Q$) as a sum of ...
0
votes
0answers
32 views

How to test if a Subset of a Ring is an Ideal.

I've been browsing Ideal test and I'm trying to understand what constitutes a concise and efficient method of testing to see if a subset is an ideal of a ring. I understand For $I$ to be an ideal of ...
2
votes
0answers
34 views

Euclidean domains and Fields

I've been wrtiting a chain of inclusions of algebraic structures as given at the end of this first paragraph on wikipedia: http://en.wikipedia.org/wiki/Euclidean_domain And I've been giving examples ...
1
vote
2answers
43 views

What are the semisimple $\mathbb{Z}$-modules?

What are the semisimple $\mathbb{Z}$-modules? Comments: I think they are direct sums of copies of such $\mathbb{Z}_p$'s, where $p$ is a prime number. I believe it is, but I can not prove.
3
votes
3answers
47 views

Integral Domains and Unique Factorisation Domains

I'm learning about Rings, commutative rings, IDs, UFDs, etc with each being a subset of the predecessor, and I'm now trying to find an ID that is not a UFD I understand $\mathbb Z[\sqrt{-5}]$ is an ...
3
votes
1answer
37 views

Ring automorphisms of $\mathbb{Q}[\sqrt{2}]$

What are the all possible ring automorphisms of $\mathbb{Q}[\sqrt{2}]$? According to me, it is completely determined by its value on $\sqrt{2}$. Am I correct?
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0answers
14 views

Let P be a proper left ideal of R. Want to show that if P is comaximal with every non zero 2 sided ideal of R, Core(P) = {0}.

Let P be a proper left ideal of R. Want to show that if P is co-maximal with every non zero 2 sided ideal of R, Core(P) = {0}. The definition I am using of comaximal is: "I is comaximal with J if ...
2
votes
1answer
38 views

How to show that $\mathbb Z+x \mathbb Q[x]$ is a GCD domain ?

How to show that $\mathbb Z+x \mathbb Q[x]$ is a Bezout domain , that is sum of two principal ideals is again a principal ideal ? Or at least , how to show that it is a GCD domain ? ( This will then ...