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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$F[x]/(p(x))\cong F$ precisely when degree of $p(x)$ is $1$

Let $F$ be a field, $p(x)\in F[x]$ an irreducible polynomial, and $F[x]/(p(x))$ be the set of equivalence classes modulo $p(x)$. I think that this is true: $F\cong F[x]/(p(x))$ precisely when the ...
2
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2answers
42 views

Can we use Eisenstein's Irreducibility Criterion to show that $x^4+1$ is not reducible in Q?

As such: Let $a(x)=x^4+1\in\mathbb{Q}\left[x\right]$. Then choose any prime $p$. By Eisenstein's Criterion, we see that $p\nmid 1$, $p\mid 0$ (since all coefficients of intermediate terms are 0), and ...
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1answer
51 views

How do I find the ideals in the ring $\mathbb F_3[x]/(x^2+2)$?

Clearly $\{0\}$ and $\mathbb F_3[x]/(x^2+2)$ will be ideals. How would I find the others?
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1answer
18 views

Multiplicative identity in a monoid ring.

Let $R$ be a ring and $S$ a subset of $R$. I want to prove that $1:S\rightarrow R: s \mapsto 1_R$ is the multiplicative identity in the ring $(R^{(s)},*,+,1,0)$ (with $R^{(S)}$ the subset of $R^S$ ...
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0answers
41 views

Prove that if R[x] is a PID, then R is a field

I just need someone to check my proof and provide me feedback: Since $R[x]$ is a PID, then the ideal $I = (x-1)$ generated by the polynomial $x-1$ is maximal because it is of degree 1 added to a ...
0
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1answer
35 views

Number of zero divisors in a finite nontrivial ring?

Let A and B be finite non trivial rings, show that the ring $A \times B$ contains at least $|A| + |B| - 2$ many zero divisors. multiplication in this case is defined as: $(a,b)\times (c,d) = (a\times ...
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0answers
28 views

zero divisors in commutative ring with unity?

Suppose A is a commutative ring with unity. prove if a,b ∈ A, with a not equal to 0, prove if a(b + 1 A ) = b(a + a) then a is a zero divisor or b = 1 A. This is what I have so far- just using ...
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1answer
14 views

If $P/I_kP$ are finitely-generated, is it true that $P/IP$ is finitely-generated where $I=\bigcap I_k$?

I'm looking into some old results on "big projectives'', and trying to understand some steps. Assume that $R$ is a (commutative) ring and $I_1,\ldots,I_n$ are ideals. Let $I=I_1\cap\cdots\cap I_n$ be ...
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2answers
62 views

Show that if $p(x)$ is reducible in $F[x]$, then $F[x]/(p(x))$ is not an integral domain.

So I know that for it to be an integral domain it has to have the following properties: Commutative Has multiplicative identity No Zero-Divisors and if $p(x)$ is reducible it can be written as ...
1
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1answer
32 views

Irreducible factorisation of polynomial over quotient field

Let $F=\mathbb{Z}_3[x]/<x^2+1>$. Factor $x^4+2$ into irreducibles in $F[x]$. I know that $F$ is a field since $x^2+1$ is irreducible. The usual way to find out that a polynomial is irreducible ...
0
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1answer
39 views

The Ring extension isomorphic to the field extension

Let $\alpha$ be algebraic over $F$, with $F(\alpha)$ the smallest field containing both $F$ and $\alpha$, and with $F[\alpha]$ the smallest ring containing both $F$ and $\alpha$. I want to show ...
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0answers
24 views

Center of matrices over a field [duplicate]

I'm trying to find the center of $\mathbb{M}_n(K)$ with $K$ a field. I know what the center would be if $K$ was a ring, but I think this isn't the same for a field $K$. In particular I'm trying to ...
1
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1answer
25 views

Is there a non-trivial ordered ring with an “integer-esque” modulo function?

(I'm inspired by this question.) Is there a [not-necessarily-commutative non-simple ordered ring with a 1 that's not equal to 0] which is not isomorphic to the integers but is such that for all ...
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0answers
28 views

The localisation of the ring $\mathbb{Z}$ at the prime ideal $(p)$ is PID [duplicate]

If $p$ is prime number, prove that $$\mathbb{Z}_{(p)}=\{\frac{a}{b}\in\mathbb{Q}: p\text{ doesn't divide }b\}$$ is a PID. So, first step is to show that $\mathbb{Z}_{(p)}$ is an Ideal Domain and ...
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1answer
14 views

Finding a prime ideal in the ring of dyadic fractions

Suppose we have the ring of dyadic fractions $R = \left\{\frac{a}{2^n} : a \in \mathbb Z, n \in \mathbb Z_{\geq 0}\right\}$. How can I find any prime ideal in this ring? I know by Krull's theorem that ...
0
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1answer
23 views

What is the name for rings that are a union of ideals and finitely many subrings?

Some rings have the property that they are a union of (proper, nontrivial) ideals. $\mathbb{Z}$ is a union of $\{\langle 2\mathbb{Z} \rangle, \langle 3\mathbb{Z} \rangle, \langle 5\mathbb{Z} \rangle ...
3
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1answer
34 views

Can somebody explain me this proof on Kasch book (Modules and rings)?

I have a question about one step in the proof of Proposition 13.2.6: If $R_R$ is injective and ${_R}R$ is noetherian then $R_R$ is a cogenerator and $R$ is artinian on both sides. In the proof of ...
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3answers
633 views

Are Euclidean domains exactly the ones which we can define “mod” on?

A Euclidean domain $E$ is an integral domain where, for any $a, b \in E$, we can write: $$a = bq + r$$ with either $r = 0$ or $f(r) < f(b)$, where $f$ is the valuation function attached to $E$. ...
1
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1answer
41 views

Being a regular embedding is an open condition for locally Noetherian schemes

Exercise $8.4.G$ of Vakil's algebraic geometry notes asks us to prove: If a locally closed embedding $\pi:X\rightarrow Y$ of locally Noetherian schemes is a regular embedding at $p$, then it is a ...
1
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1answer
59 views

$\mathrm{Frac}(R)\subseteq\mathrm{Frac}(S)$ algebraic implies $R\subseteq S$ algebraic?

EDIT: The matter is now resolved. Here is the statement and the proof. Theorem: Let $R\subseteq S$ be an extension of integral domains and let $K\subseteq L$ be the corresponding fields of fractions. ...
1
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1answer
44 views

I don't understand a step of the proof that a nonzero finite commutative ring with no zero divisors is a field.

This seems to be a fairly classic problem in algebra: Let $R$ be a nonzero finite commutative ring with no zero divisors. Prove that $R$ is a field. Here is the a solution I came across: It has ...
0
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1answer
30 views

Find a prime $p>5$ such that $x^2 +1$ is reducible in $\mathbb Z_p[x]$

Find a prime $p>5$ such that $x^2 +1$ is reducible in $\mathbb Z_p[x]$. Can anyone please give me some hints as to how I can go about finding this value of $p$?
3
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3answers
76 views

Is there a ring homomorphism between $\mathbb{Z}$ into $\mathbb{Z} \times \mathbb{Z}$?

Is there a ring homomorphism between $\mathbb{Z}$ into $\mathbb{Z} \times \mathbb{Z}$? Also for any rings $R_{1}$ and $R_{2}$ does there exist a ring homomorphism $\phi$ : $R_{1} \rightarrow$ $R_{1} ...
0
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1answer
26 views

What is this ideal equal to? What is it called? “composition ideal in $R[X]$”

Let $R$ be a ring and $f(X)=f_0+f_1X+\dots +f_n X^n\in R[X]$. Define $f(J) \equiv f_0 + f_1 J + \dots + f_n J^n$ where $J^k$ is the $k$th power ideal, and $A + B = \{a + b : a \in A, b \in B\}$. ...
0
votes
1answer
39 views

Euclidean evaluation

Let $R$ be a commutative ring and let $a, b \in R\setminus\{0\}$. Suppose that $d$ is a gcd of $a$ and $b$. Suppose further that $R$ is a Euclidean domain with valuation $\delta$. Let $c$ be another ...
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2answers
40 views

Showing that $\langle x,y\rangle^2 \subsetneq \langle x,y^2\rangle$

I've the ring $K[x,y]$, where $K$ is a field. How should I show that the ideal $\mathfrak q=\langle x,y^2\rangle$ contains ideal $\mathfrak p^2$ properly, where $\mathfrak p=\langle x,y\rangle$. ...
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0answers
34 views

Is there an accepted definition of coprimality in commutative ring theory?

I can think of at least three possible definitions of coprimality in commutative ring theory: $a,b \in R$ are coprime iff if $c \mid a$ and $c \mid b$, then $c \mid 1$. if $a \mid c$ and $b \mid c$, ...
0
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1answer
26 views

The order ideal of a finitely generated module over a Dedekind domain

Let $R$ be a Dedekind domain and $M$ be a nonzero finitely generated torsion $R$-module. In Curtis and Reiner's Methods of Representation Theory it states that $M$ has a composition series and if its ...
0
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1answer
49 views

Assume that $I/J$ is a prime ideal of $R/J$, is $I$ a prime ideal of $R$? [on hold]

$R$ is a commutative ring. $I$ and $J$ are ideals of $R$ with $J\subseteq I$. Assume that in the quotient ring $R/J$, an ideal $I/J$ is prime ideal. Is the ideal $I$ in $R$ is prime ideal?
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2answers
82 views

How many homomorphisms are there of $\Bbb Z \times \Bbb Z\times \Bbb Z$ into $\Bbb Z$?

I've tried looking around for an explanation to this problem, but I've been having trouble finding a clear solution that specifically focuses on this question: How many homomorphisms are there of ...
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2answers
42 views

What does the “set R- {-1}” mean??? [closed]

I am trying to answer the question: On the set R-{-1} define the operations a⊕b=a+b+aba⊕b=a+b+ab and axb=0axb=0. Determine if (R - {-1}, ⊕, x) is a ring. Is it a commutative ring with unity? However ...
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1answer
54 views

Determine if R is a commutative ring with unity?

On the set $R-\{-1\}$ define the operations $a\oplus b = a + b + ab$ and $a \times b = 0$. Determine if $\big(R-\{-1\}, \oplus,\times\big)$ is a ring. Is it a commutative ring with unity? Using the ...
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3answers
66 views

How can I show that any commutative ring R has a maximal ideal?

Suppose I have a commutative ring R and some ideal in R, say I. I know that I will have to be a two-sided ideal since R is commutative. I know I will have to come up with some sort of Zorn's lemma ...
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1answer
49 views

If $x$ is not nilpotent, how to prove there exists a prime ideal does't contain $x$

Actually, I find some explanation using Zorn's lemma and localization. However, our class doesn't include these until now. So can someone prove it in an easier way?
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1answer
46 views

ideal in ring R is subgroup of additive group $R^+$

I was told that any ideal in a ring $R$ is a subgroup of the additive group $R^+$, I was also told that it is actually a normal subgroup. I am a bit skeptical about this, but cannot think of any ...
2
votes
2answers
149 views

Prove that additive order is preserved by isomorphisms

I want to show that $\mathbb{Z}_4$ and $\mathbb{Z}_2 \times \mathbb{Z}_2$ are not isomorphic by using the fact that $\mathbb{Z}_4$ has one element of additive order 4 (the largest additive order), ...
2
votes
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 ...
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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 ...
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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
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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
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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
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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 ...
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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
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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
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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
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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
164 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 ...