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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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 ...
-1
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0answers
26 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 ...
0
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2answers
42 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
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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
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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
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2answers
34 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 ...
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1answer
36 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. ...
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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
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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
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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 ?
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1answer
53 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?
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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
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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
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1answer
48 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
39 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 $x,y \in R$.
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1answer
51 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 $R$.
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2answers
46 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 ...
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1answer
24 views

Let R = Z[ √ −7] = {a + b √ −7 | a, b ∈ Z} [closed]

Let R = Z[√−7] = {a + b√−7 | a, b ∈ Z}. Note that R is a subring of C, which is a field, so R is an integral domain. Let the function N : R → Z be defined by N(a + b√−7) = a^2 + 7b^2 (a) Prove that ...
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votes
1answer
26 views

Using the first isomorphism theorem, find a quotient ring of $R[x]$ which is isomorphic to $R[\sqrt{−3}]$ [closed]

Consider the ring homomorphism $\phi : R[x] → R[\sqrt{−3}]$ defined by $\phi(x) = \sqrt{−3}$. Using the first isomorphism theorem, find a quotient ring of $R[x]$ which is isomorphic to ...
1
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1answer
87 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 $(M_2(\Bbb Z_3))^*$.
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2answers
26 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 ...
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3answers
46 views

$R = \Bbb F_7[x]/(x^2+2)$. Is $R$ a field? [closed]

$R = \Bbb F_7[x]/(h)$ where $h(x) = x^2 + 2$. Is $R$ a field? justify with examples. please help!
2
votes
1answer
38 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
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1answer
50 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
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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
35 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 ...
1
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2answers
42 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
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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 ...
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2answers
159 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
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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
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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 ...
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2answers
42 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.
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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 ...
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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 ...
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1answer
33 views

Looking for an example of a GCD domain which is not UFD

I know that every UFD (unique factorization domain ) is a GCD domain i.e. g.c.d. of any two elements , not both zero , exists in the domain . I am looking for an example of a GCD domain which is not ...
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2answers
41 views

Prime Ideals and multiplicative sets

I am currently studying a course on commutative algebra and came across this statement: An Ideal $I$ in a ring $R$ is prime if and only if $R\setminus I$ is a multiplicative set. I have proved ...
1
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3answers
59 views

Show that $R/(I \cap J) \cong (R/I) \times (R/J) $

My question actually follows from this one: Show that if $I + J = R$, then $R/(I \cap J) \cong R/I \times R/J$ What I don't understand is why is it necessary for $I+J=R$, in order for $$ ...
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2answers
30 views

Why is the Jacobson radical of the integers {0}.

Why is the Jacobson radical of the integers {0}? I have been working through questions dealing with the Jacobson radical and have come across this and can't think of why this would be. Any help ...
2
votes
1answer
84 views

When is $G\cong\operatorname{End}(G)$?

$\newcommand\End{\operatorname{End}}$Let $G$ be an Abelian group. Are there sufficient conditions for the existence of an isomorphism $G\cong\End(G)$, where $\End(G)$ is considered a group under ...
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1answer
52 views

The relation between prime ideal and simple ring

I saw "Any simple ring is a prime ring" as an example in Prime ring@wiki. Can anyone show me how to proof it? Also, on the other side, is any prime ring a simple ring? Thanks.
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4answers
71 views

All prime ideals are maximal - Counterexample

I would like to know of some simple counter examples to the statement "ALL prime ideals are maximal" I say counter examples because I think the statement isn't true.
2
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4answers
147 views

$x$ is a left zero-divisor $\iff$ $x$ is a right zero-divisor.

Let $R$ be a ring with unity. Show that $x$ is a left zero-divisor if and only if $x$ is a right zero-divisor. Suppose, $x$ is a left zero divisor. Then, $\exists y \in R$ such that $xy = 0 ...
3
votes
1answer
31 views

Further examples of Principal Ideal Domain that are not Euclidean Domains

In several courses of algebra, I've heard that not all PIDs are EDs, and the canonical example is $\mathbb{Z}\left[\dfrac{1+\sqrt{-19}}{2}\right]$ which I've heard over and over. Some cursory research ...
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1answer
35 views

Show that if $M$ is a semisimple artinian module then $M$ is finitely generated.

The exercise is as follows: Show that for a semisimple module $M$ over any ring, the following conditions are equivalent: $(1)$ $M$ is finitely generated; $(2)$ $M$ is Noetherian; $(3)$ $M$ is ...