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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Division Rings and Division Algebras

On wikipedia (division algebra page), it says "Wedderburn's little theorem states that if D is a finite division algebra, then D is a finite field." I've just been through a proof this theorem, but ...
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2answers
86 views

Is a set that is an abelian group under addition and a group under multipliation a field?

I suspect the answer to my question is yes, but I'm just checking my understanding. If we have a set which is an Abelian group under addition and a group under multiplication is it then defined as a ...
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2answers
129 views

What is the ring $\Bbb Z[x]/(x^2-3,2x+4)$? [duplicate]

How to describe an element of the Ring $\Bbb Z[x]/(x^2-3,2x+4)$? Or is it isomorphic to any well known ring? What is meant by quotienting $\Bbb Z[x]$ by a kernel like that?
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1answer
254 views

An example of ideal that has no primary decomposition.

Give an example of a commutative ring with unit and an ideal that has no primary decomposition. I think boolean Ring will be the right example, but I don't know how I must show that. So please ...
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2answers
53 views

Factor ring of polynomial

$F[x]$ is a polynomial ring over a certain field $F$. $J$ is an ideal of $F$, $J = (f(x))$. I need to prove that if the polynomial $f(x)$ has a multiple root the factor ring $F[x]/J$ is not a field. ...
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1answer
126 views

Prove a (unital) ring with this property has no zero divisors?

Let $\mathcal{R}$ be a unital ring with the property that $\forall p \not= 0_{R}$, $\exists$ unique $q$ such that $p = pqp$. Prove that $\mathcal{R}$ has no zero divisors. These are some potentially ...
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0answers
123 views

Ideals of infinite product rings

(Rewritten following earlier feedback...) (1) What are the ideals of ${\mathbb Z}^{\mathbb N}$? That is, take the ring which is the product of a countable infinity of copies of $\mathbb Z$; is ...
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1answer
55 views

For a Euclidean Domain, prove that $\delta(z)>0$ if $z\in R$ is not a unit.

For a Euclidean Domain, prove that $\delta(z)>0$ if $z\in R$ is not a unit, where $\delta$ is the Euclidean function. Is it just since $z$ is not a unit then $\delta(z)>\delta(1)>0?$ Please ...
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2answers
62 views

Inverse image of a PID is a PID

Let $f : R \to S$ be a ring homomorphism from $R$ onto $S$. If $S$ is a PID, is $R$ then a PID? If this is not possbile, is there an example to contradict it?
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1answer
403 views

Describe the units in $\mathbb{Z}[\sqrt{-d}]$

I need help describing all the units in the ring $\mathbb{Z}[\sqrt{-d}]=\{a+b\sqrt{-d}|a,b\in\mathbb{Z}\}$ where $d>0$ and is square-free. Thanks.
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1answer
188 views

$\mathbb{Z}[\sqrt{-19}]$ is not a PID? Proof correction?

Prove or disprove the following: The ring $\mathbb{Z}[\sqrt{-19}]$ is a PID. Going over some questions from a course I took recently, I want to see if I am understanding this problem well. My ...
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0answers
40 views

integral domain with a field as a subring [duplicate]

I would like to know if my solution to the following exercise is correct. Let $A$ be an integral domain (with a unit) which has a field $\mathbb K$ as a subring and such that $A$ is a ...
3
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1answer
491 views

Find maximal ideals of a ring

I'm trying to find all the maximal ideal of the ring $\mathbb{Z}[\sqrt{3}] = \{a+b\sqrt{3} : a,b\in \mathbb{Z}\}$. I have found one, that is $A = \{3a + b\sqrt{3} : a,b \in \mathbb{Z} \} $, and I ...
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1answer
62 views

Number of unitary homomorphisms $\phi \ : \ \mathbb{Z}[X]/(X^3+3X+5) \longrightarrow \mathbb{R}$

I need some help to solve an exercise: What is the number of unitary homomorphisms $\phi \ : \ \mathbb{Z}[X]/(X^3+3X+5) \longrightarrow \mathbb{R}$?. Research effort The zeros of $X^3+3X + 5$ must ...
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0answers
64 views

Prove that this factor ring is a finite ring without zero divisors [duplicate]

Let $R=\mathbb{Z}[\sqrt{d}]=\{a+b\sqrt{d}\mid a,b,d\in\mathbb{Z}\}$ and let $d$ be square-free. Let $P$ be a non-zero prime ideal in $R$. I need to prove that the factor ring $R/P$ has no zero ...
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1answer
30 views

Showing $f(x^{p_1}) \mid f(x^{p_1 p_2})$ Given that $f(x) \mid f(x^{p_1}), f(x^{p_2})$

Hypothesis: Let $f = a_0 + a_1x + \ldots + a_nx^n \in \mathbb{Z}[x]$. Suppose $f(x) \mid f(x^{p_1})$ and $f \mid f(x^{p_2})$ for $p_1$ and $p_2$ two positive prime integers. Goal: Show that ...
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2answers
76 views

Ring isomorphism, Unity is preserved

I am wondering that if I have a ring isomorphism, /phi going from R to R', where R is a ring with unity, how can I prove that R' is also a ring with unity? It seems to be very obvious so I don't know ...
3
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1answer
24 views

Finding an $i \in \mathbb{N}$ s.t. $im + k = p$

Let $m,k \in \mathbb{N}$ s.t. $\gcd(m,k) = 1$. Let $\pi$ be a prime positive integer. Question: Does there always exist an $i \in \mathbb{N}$ s.t. $im + k = p$ s.t. $p$ is a prime positive integer ...
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3answers
137 views

Minimal polynomial of $\alpha^2$ given the minimal polynomial of $\alpha$

Given that $\alpha$ is a root (in the field extension) of the irreducible polynomial $X^4+X^3-X+2\in\mathbb{Q}[X]$, I have to find the minimal polynomial of $\alpha^2$. I am thinking about this for a ...
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1answer
33 views

If $a/s$ is irreducible in $R_{S}$, then $a$ is irreducible in $R$

Let $R$ be a UFD, and let $S$ be a multiplicative subset of $R$ containing the unity of $R$. Show that if $a/s$ is irreducible in $R_{S}$, then $a$ is irreducible in $R$. I showed that if $a$ is ...
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2answers
54 views

Find $c_1,c_2,c_3\in\mathbb{Q}$ such that $(1+\alpha^4)^{-1}=c_1+c_2\alpha+c_3\alpha^2$ in $\Bbb Q(\alpha)$.

Let $\alpha\in \overline{\mathbb{Q}}$ be a root of $X^3+X+1\in\mathbb{Q}[X]$. So this is the minimal polynomial of $\alpha$ because it's irreducible in $\mathbb{Q}[X]$. I had to find the minimal ...
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1answer
94 views

Is $\mathbb{Z}[x]/(x^2+1)$ isomorphic to $\mathbb{Z}[i]$?

Is $\mathbb{Z}[x]/(x^2+1)$ isomorphic to $\mathbb{Z}[i]$? My attempt is that try to define a mapping $g$ from $\mathbb{Z}[x]$ to $\mathbb{Z}[i]$ by $g(f(x))= f(i)$, for $f(x)\in\mathbb{Z}[x]$. ...
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2answers
101 views

Maximal ideals in $C^\infty(\mathbb{R})$

I know that for a compact manifold $M$ any maximal ideal in the algebra $C^\infty(M)$ of smooth functions on $M$ is of the form $\mathfrak{m}_p=\{f\in C^\infty(M)|f(p)=0\}$. For example, the proof is ...
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1answer
79 views

$\mathbb Z[i]$ and the ideal $(5)$ [closed]

Consider $\mathbb Z[i]$ the ring of Gaussian integers and its ideal $J=(5)$. Show that $\mathbb Z[i]/J \cong \mathbb Z_5 \oplus \mathbb Z_5$ as rings.
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5answers
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Why is ring addition commutative?

What is the motivation behind axiomatically forcing the underpinning group of a ring to be abelian? Noncommutative rings are vastly more complex than commutative ones, so I am assuming that allowing ...
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1answer
111 views

Showing something is a maximal ideal.

Let $\varphi:R\to S$ be an onto ring homomorphism. Suppose $M$ is a maximal ideal of $R$, and let $N=\{\varphi(m)|m\in M\}$. Show that $N$ is a maximal ideal of $S$. Assume that $N$ is an ideal of ...
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2answers
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whether $x^4+x^3+x^2+x+1$ is irreducible over $\mathbb Z$

How to check whether $x^4+x^3+x^2+x+1$ is irreducible over $\mathbb Z?$ My guess: If $x^4+x^3+x^2+x+1$ is reducible then $$x^4+x^3+x^2+x+1=f(x)g(x)$$ where $1\le\deg f(x),\deg g(x)\le3$ and $\deg ...
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1answer
48 views

Powers of elements in rings

Boolean rings are examples of rings where for any $x$ there exists $n>1$ such that $x^n=x$. (Of course $n=2$ will do.) I would appreciate some other examples.
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0answers
39 views

Finding irreducible polynomial

1, 3 False: $f_{2^2}(x)=f_4(x)=x^3+x^2+x+1=(x^2+1)(x+1)$ is not irreducible. 2 True: Cyclotomic polynomial of order prime. I am not sure about 4. Here's my guess about 4: $x^{p^{e-1}}$ is ...
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1answer
78 views

Show that $\dfrac{\mathbb Z[x]}{(2,x)}$ is a field.

Show that $\dfrac{\mathbb Z[x]}{(2,x)}$ is a field, where $$(2,x)=\{p(x)\in\mathbb Z[x]:\text{the constant term of $p(x)$ is even}\}$$ Thus $\dfrac{\mathbb Z[x]}{(2,x)}=\{(2,x),1+(2,x)\}.$ Since ...
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2answers
37 views

Which are integral domain

I know that $\mathbb Z[i]/n\mathbb Z[i]$ is an integral domain $\iff \langle n\rangle =n\mathbb Z[i]$ is a prime ideal of $\mathbb Z[i]\iff n$ is an prime element of $\mathbb Z[i].$ $2=(1+i)(1-i)$ ...
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1answer
95 views

Showing $\{a+b\sqrt{2} \in R$ | $a$ is divisible by $2\}$ is an ideal.

This was a problem in my math book and I was wondering what the proof looks like: Let $R = \{a + b\sqrt{2} | a,b\in \mathbb{Z}\}$. Show that $I = \{a+b\sqrt{2}\in R | a$ is divisible by $2\}$ is an ...
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1answer
80 views

If every ideal is principal show the same is true for the image of a homomorphism.

I was working on a worksheet for class and came across this problem: Let $\varphi:R\to S$ be an onto ring homomorphism with the property that every ideal of $R$ is principal. Show that the same ...
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3answers
74 views

Let $F$ be a field. $\langle X,Y\rangle$ is a maximal ideal of $F[X,Y]$.

Let $F$ be a field. Prove that $\langle X,Y\rangle$ is a maximal ideal of $F[X,Y]$.
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2answers
552 views

Specific way of showing $\Bbb Z[\sqrt{-d}]$ is not a Euclidean Domain when $d>2$

Is it true that if a ring is not a UFD then it's not a Euclidean Domain? I have a ring $R=\mathbb{Z}[\sqrt{-d}]=\{ a+b\sqrt{-d} \mid a,b \in \mathbb{Z} \}$ where $d$ is a square free integer. I want ...
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2answers
906 views

Prove that every nonzero prime ideal is maximal in $\mathbb{Z}[\sqrt{d}]$

$d \in \mathbb{Z}$ is a square-free integer ($d \ne 1$, and $d$ has no factors of the form $c^2$ except $c = \pm 1$), and let $R=\mathbb{Z}[\sqrt{d}]= \{ a+b\sqrt{d} \mid a,b \in \mathbb{Z} \}$. ...
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3answers
153 views

Products of ideals is an ideal and comaximal ideals

Let $I$ and $J$ be two ideals in a ring $R$. Prove that $IJ$ is an ideal. Prove that if $R$ is a commutative ring with two ideals satisfying $I+J=R$ then $IJ=I\cap J$. I could prove that $IJ$ has ...
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2answers
618 views

Nilpotent/invertible polynomial over commutative ring. [duplicate]

Let $p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a polynomial over a commutative ring $R$. Prove that (a) $p$ is unit in $R[x]$ iff $a_0$ is unit and $a_1,a_2,\ldots,a_n$ are nilpotent in ...
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1answer
75 views

What is the splitting field of $X^{20}-1$ over $\Bbb F_3$. And how to factor $X^{20}-1$ in $\Bbb F_3[X]$

I'm doing some exercises to prepare for my exam: What is the splitting field of $X^{20}-1$ over $\Bbb F_3$. And how to factor $X^{20}-1$ in $\Bbb F_3[X]$. I've no idea how to tackle this ...
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2answers
76 views

Nontrivial ideals of integral domain intersect nontrivially

Let $R$ be an integral domain, and $A , B$ be non trivial ideals of $R$. Then prove that $|A \cap B|>1$.
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2answers
211 views

I know that Every division ring is simple. But is the converse true?

I know that Every division ring is simple. But is the converse true? I guess it is not true. But cant find a counterexample.
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348 views

Existence of prime ideals in rings without identity

Let $R$ be a commutative ring (not necessarily containing $1$). Say that $R$ is the trivial ring if it has trivial (zero) multiplication. If $R$ is the trivial ring, then $R$ has no prime ideals (as ...
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2answers
313 views

Proof of Fermats Last Theorem for Given Exponent

Where can I find reasonably short and elementary proofs (using basic concepts of ring, field, galois theory) of Fermat's Last Theorem for specific n? For example, $n=5,7,13$?
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1answer
77 views

Free modules and free submodules. [duplicate]

Just finished my semester in advanced abstract algebra and there was one question that I could not answer. Let $R$ be commutative and let $(e_1,...,e_n)$ be a base for $R^{(n)}$. Put ...
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2answers
195 views

Proving field of fractions

Basically, I'm trying to show that the field of formal Laurent series is the field of fractions for the ring of power series. My question here is, how exactly does one prove that a given field is a ...
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2answers
49 views

Frobenius maps exist/do not exist for integers?

Does there exist an infinite ring $R$ such that $(x+y)^b=x^b+y^b$, and similarly for $2$ other odd primes $a,c$; in which $\Bbb{Z}$ can be embedded as a ring? I have no idea where to begin. Maybe ...
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1answer
39 views

Does there exist a infinite ring in which there are $3$ Frobenius homs?

Does there exist an infinite ring $R$ such that $(x + y)^b = x^b + y^b$, and similarly for $2$ other odd primes $a,c$? Or what's the best that can be done?
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2answers
50 views

Extended ideals

If $R\subset S$ is a ring extension where $1\in R$ and $I$ is an ideal of $R$ is it true that $IS$, the subset of $S$ generated by $I$ is an ideal of $S$? Should we assume $R$ is commutative?
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3answers
90 views

Showing that an integral domain is a PID if it satisfies two conditions

This is just a textbook problem from Dummit and Foote, but the issue is that our class barely touched on PIDs and the preceding material, so I don't really know or understand much. Anyway, Let ...
2
votes
1answer
328 views

Ring of formal Laurent series: units and defining operations

I'm struggling to see how one could show that the ring of formal Laurent series in $F$ is unital. I defined addition and multiplication to be, for $p, q \in F((t))$: if $$p = \sum_{i \in X} a_i ...