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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1answer
21 views

Smallest Ideal containing $a$ in commutative ring without unity

Let $R$ be a commutative ring without unity such that $a$ belongs to $R$. Describe the smallest ideal containing $a$. We know that in a commutative ring with unity $(a) = ${$r.a$ | $r$ belongs to ...
1
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1answer
33 views

If $R$ is an integral domain, show that there is no subfield $k$ of Frac($R$) containing $R$.

If $R$ is an integral domain, show that there is no subfield $k$ of Frac($R$) containing $R$. I think the way to prove this is by contradiction. So, let $R$ be an integral domain, and let $k$ be a ...
5
votes
1answer
106 views

A prime ideal $\mathfrak{p}$ decomposes in $\mathbb{Q}(\zeta_{12})/\mathbb{Q}(i)$ iff it is generated by $\alpha\in1+3\Bbb{Z}[i]$

Prove that for a nonzero prime ideal $\mathfrak{p}$ of $\mathbb{Z}[i]$ which does not divide $3$, $\mathfrak{p}$ decomposes completely in the quadratic extension ...
2
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2answers
34 views

The characteristic of a direct sum of two rings

I had a query regarding the characteristic of the direct sum of two rings. That is, given two rings R and S, with characteristics m and n respectively, show that: $$char(R \oplus S)=lcm(m,n)$$ That ...
-1
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1answer
29 views

Does UFD $R$ wrap around its ideal $I$ infinitely and surjectively as $\Bbb{Z}$ does $(n)$?

If $R$ is a UFD, and $I$ is an ideal of $R$, then do elements of $R$ wrap around $I$ as they do in the case of $\Bbb{Z}$ and $(n)$. And by that I mean, letting $\pi : R \to R / I$ be the natural ...
0
votes
2answers
69 views

Noetherian submodules and isomorphisms

Suppose $M$ is an $A$-Module, and $N$ is a submodule of $M$. Let $f:N\to M$ be an $A$-module epimorphism. How could I show that if $N$ is noetherian, then $f$ is an isomorphism? Thanks
2
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0answers
49 views

Is $(2, \sqrt{m})$ a principal ideal or not in the ring $\mathbb{Z}[\sqrt{m}]$ [closed]

Let $m$ be a negative even integer. In the ring $\mathbb{Z}[\sqrt{m}]$, is the ideal $(2,\sqrt{m})$ a principal ideal?
1
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2answers
80 views

Factor 65 into irreducible in $\mathbb{Z}[i]$

Factor 65 into irreducible in $\mathbb{Z}[i]$ I tried to factor 65 in Gaussian integers by Mathematica, and I got $65 = -(1+2i)(2+i)(2+3i)(3+2i)$, but i don't know how to factor it by hand. Could you ...
6
votes
2answers
91 views

Describe all ring homomorphisms from $\mathbb{R}[T] \rightarrow \mathbb{R}[T]$

One of the problems in a problem set I was given as homework in my Algebra course proposes the next problem: Describe all ring homomorphisms $\mathbb{R}[T] \rightarrow \mathbb{R}[T]$. Which of ...
1
vote
1answer
28 views

Valuation ring and integral closure

Let $A$ be a one-dimension local noetherian domain and suppose that we know that $K=\text{Frac}(A)$ is a complete discrete valuation field (valuations for me are surjective). Let's denote with ...
4
votes
1answer
40 views

Descending chain condition and an identity

Let $R$ be a ring with descending chain condition on right ideals. Suppose $l(R)=0$ (the left annihilator of $R$) and $\exists c\in R$ with $r(c)=0$ (the right annihilator of $c$). Show that $R$ has ...
4
votes
2answers
74 views

Is the tensor product of non-commutative algebras a colimit?

For $R$ a commutative ring, the tensor product of $R$-algebras is the coproduct in the category of commutative $R$-algebras. In the noncommutative case it is no longer the coproduct in the category of ...
0
votes
1answer
82 views

Prove that $\langle\sqrt2\rangle$ is a maximal ideal in $\Bbb Z[\sqrt2]$. How many elements are in $\Bbb Z[\sqrt2]/\langle\sqrt2\rangle$? [closed]

Prove that $\langle\sqrt{2}\rangle$ is a maximal ideal in $\Bbb Z[\sqrt{2}]$. How many elements are in the ring $\Bbb Z[\sqrt{2}]/\langle\sqrt{2}\rangle$ ? I am unable to solve this. Please help ...
2
votes
0answers
73 views

Algebraic over algebraic is algebraic?

Let $R_1\subseteq R_2\subseteq R_3$ be integral domains. I want to show that if $R_3$ is algebraic over $R_2$ and $R_2$ is algebraic over $R_1$ then $R_3$ is algebraic over $R_1$. [To emphasize: this ...
2
votes
1answer
20 views

Local subring of a DVR and finite residue field extension

Let $\mathcal O$ be a complete DVR with fraction field $K$, maximal ideal $\mathfrak p$ and residue field $\widetilde K=\mathcal O/\mathfrak p$. Now consider a subring $A\subset \mathcal O$ with the ...
3
votes
1answer
47 views

Nilpotent or just nil idempotent ideal?

Let $R$ be a ring with zero Krull dimension and $I$ be an idempotent ideal contained in the Jacobson radical $J(R)$ of $R$. Could one infer just with these hypotheses that $I$ is a nilpotent ideal? I ...
0
votes
0answers
21 views

Ring of all $\mathbb{C}$-valued continuous functions on closed interval Noetherian? [duplicate]

So I was trying to construct examples of rings that weren't Noetherian, and here was one I was not too sure about. Let $R$ be the ring of all $\mathbb{C}$-valued continuous functions on the closed ...
1
vote
0answers
23 views

Injective $A$-homomorphism is also surjective? [duplicate]

Let $A$ be a ring and let $M$ be an Artinian module over $A$. Let $f: M \to M$ be an $A$-homomorphism. Assume $f$ is injective. Does it follow that $f$ is surjective? I'm inclined to think yes, and ...
5
votes
2answers
52 views

S,R bimodules subcategory of the category of S+R modules?

If $R$ and $S$ are commutative rings, then does the category $R \oplus S$-modules encompase the category of $(S,R)$-bimodules? I was thinking we can accomplish this by defining the action to be: ...
0
votes
0answers
37 views

Gauss integers and euclidean division

Does someone know if there's a way to show that the Gauss integers are an euclidean ring without using the complex numbers (except for the norm, needed in the definition of euclidean ring), only by ...
0
votes
1answer
26 views

Can I conclude that the following map is surjective?

I have a module homomorphism $A \rightarrow B \oplus C $ whose projection onto the first factor $B$ is surjective. If the projection onto $C$ is surjective and $C$ is a simple module, can I conclude ...
1
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1answer
30 views

Projective modules over a direct product of rings

Let $R$ and $S$ be rings, and let $\text{proj}(R)$ denote the category of finitely generated projective modules. Is there an equivalence of categories between $\text{proj}(R \times S)$ and ...
0
votes
2answers
57 views

Chain of inclusions between classes of rings [closed]

I need to show using a chain of inclusions the relationship between the types of rings which are = {rings, commutative rings, integral domains, Euclidean domains, principal ideal domains, unique ...
4
votes
1answer
65 views

Does the prime ideal $(p)$ of $\mathbb{Z}[\sqrt{-5}]$ split completely in the extension of $\mathbb{Q}(\sqrt{-5}, i)/\mathbb{Q}(\sqrt{-5})$?

For prime numbers $p$ such that $p \equiv 11$, $13$, $17$, $19 \text{ mod }20$, does the prime ideal $(p)$ of $\mathbb{Z}[\sqrt{-5}]$ split completely in the extension of $\mathbb{Q}(\sqrt{-5}, ...
1
vote
2answers
64 views

Is a subring a UFD? [duplicate]

If $K$ is a field is $K[X^2,X^3]$ a UFD when considered as a subring of $K[X]$? I know that $K[X]$ is a $PID$ but nothing else ,these ring theory question just dont seem to occur to me..please ...
0
votes
1answer
70 views

Quotient ring is a domain??

i wanted to ask this ring theory related question My question is, is $\frac {\mathbb{C}[X,Y]}{(X^4+X^3Y+Y^4)}$ a domain or not... i know that for it to be a domain the denominator ideal has to be ...
0
votes
1answer
47 views

Calculate the spectrum of two given quotient rings

I am trying to do the following exercise: Let $K$ be a field. Calculate the spectrum of the following rings (1) $K[X] / \langle X^2 \rangle$ (2) $K[X]/ \langle X^2(X+1) \rangle$ For the first ...
0
votes
2answers
67 views

Radical of the powers of an ideal

I am asked to prove the following: $$\sqrt{\mathfrak{a}^n} = \sqrt{\mathfrak{a}}$$ Here is my attempt so far: $\sqrt{\mathfrak{a}^n} \subseteq \sqrt{\mathfrak{a}}:$ (By Induction) Clearly the ...
3
votes
4answers
71 views

A unit in $\mathbb{Z}_{9}[x]$

Show that $1+3x$ is a unit in $\mathbb Z_9[x]$. Hence corollary 4.5 may be false if $R$ is not an integral domain. The corollary 4.5 is: Let $R$ be an integral domain and $f(x) \in R[x]$. Then ...
1
vote
2answers
56 views

Unnatural homomorphism form domain $R$ to $Frac (R)$

There is a natural homomorphism for $R$ to $Frac (R)$ that sends $r\rightarrow(r,1)$, but beside this injective homomorphism, is there example of ring $R$ s.t there exist another injective ...
1
vote
3answers
100 views

the ideal contains some power of the unique prime ideal containing it

Let $R$ be a Noetherian ring, and $I$ an ideal such that there exists a unique prime ideal $\mathfrak{p}$ containing $I$. Show that $I$ contains some power of $\mathfrak{p}.$ In this question, I ...
2
votes
2answers
54 views

ring isomorphism from $\mathbb Z_3\times \mathbb Z_2$ to $\mathbb Z_6$

Problem is ring isomorphism from $\mathbb Z_3\times \mathbb Z_2$ to $\mathbb Z_6$. My solution: Let $f=\mathbb Z_3\times \mathbb Z_2 \to \mathbb Z_6$ defined as $f(a_3,b_2)=ab_6$. I prove it is ...
0
votes
1answer
30 views

Order of an element in an integral domain

Suppose that (R, + , •), is an integral domain, where + and • are the usual operations, addition and multiplication respectively, and the non zero element r, as considered as an element of the abelian ...
2
votes
1answer
40 views

Show that the Group Ring $F_p[G]$ where $G$ is a $p$-Group has a unique maximal ideal.

Show that the Group Ring $F_p[G]$ where $F_p$ is finite field of order $p$ and $G$ is a $p$-Group (not necessarily abelian) has a unique maximal ideal, i.e. it is a local ring. Attempt: Consider ...
0
votes
2answers
42 views

If $I,J$ are ideals in a polynomial ring over a field, how do I prove that $I = J$ if $\operatorname{in}_<(I)=\operatorname{in}_<(J)$?

If $I\subseteq J$ are ideals in a polynomial ring of $n$ variables, how do I prove that $I = J$ if $\operatorname{in}_{\lt}(I)=\operatorname{in}_{\lt}(J)$, where $\lt$ is any monomial ordering? ...
3
votes
1answer
35 views

completion and heights of prime ideals

Let $A$ be a noetherian, regular local domain of dimension $2$ (for instance the local ring at a smooth point of a surface) and consider its completion $\hat A$ at its maximal ideal. Now let's look at ...
0
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0answers
63 views

When is a map $ f : A^{n+1} \to A^n $ injective? [duplicate]

Is there an example of a commutative ring $ A $ with unit $ 1_A $ and an $A$-module map : $ f : A^{n+1} \to A^n $ such that $f$ is injective ? If the answer is yes, when is this enunciation false ? ...
0
votes
0answers
49 views

Prime element and irreducible

I would like to know whether a polynomial in $\mathbb Z[x]$ is a prime element if and only if it is irreducible. Since $\mathbb Z[x]$ is an integral domain, a prime element in $\mathbb Z[x]$ is ...
1
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1answer
75 views

Contraction of an ideal

Let $f: \mathbb{Z}[X] \longrightarrow \mathbb{Z}[\sqrt{2}]$ be a ring homomorphism sending $X$ to $\sqrt{2}$. I am asked to compute a few contractions, and I am wondering if I could get some help ...
0
votes
1answer
49 views

prove $\mathbb{Z}(G\times C_2)\cong (\mathbb{Z}G)C_2$

I want to prove $\mathbb{Z}(G\times C_2)\cong (\mathbb{Z}G)C_2$, where $C_2=<x| x^2=1>$, where $\mathbb{Z}(G\times C_2)$ and $(\mathbb{Z}G)C_2$ are integral group rings and I am looking for ring ...
2
votes
0answers
45 views

Existence of minimal prime ideal contained in given prime ideal and containing a given subset

Let $R$ be a unital commutative ring, $P$ $\subseteq$ $R$ a prime ideal, $X\subseteq P$ a subset. Show there exists a minimal (inclusion minimal) prime ideal contained in $P$ which contains $X$. ...
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votes
7answers
497 views

Morphisms in the category of rings

We know that in the category of (unitary) rings, $\mathbb{Z}$ is the itinial object, i.e. it is the only ring such that for each ring $A,$ there exists a unique ring homomorphism $f:\mathbb{Z} \to A$. ...
1
vote
1answer
30 views

Show that for a Noetherian ring R, the R-module R$^{n}$ is Noetherian

I'm allowed to use this: An R-module M is Noetherian when a submodule N is Noetherian and their quotient M/N is Noetherian. In my particular problem, M = R$^{n}$ and N = R. I believe I will have ...
1
vote
1answer
32 views

Relation between singular homology and cohomology

Is it true that for every $\sigma \in H_k(X;R)$ there is a $\phi \in H^k(X;R)$ such that $\phi(\sigma) = 1$ for every topological space $X$ and ring with unity $R$ ? That strikes me as a very ...
0
votes
2answers
17 views

showing that an R-module is Noetherian if a submodule is Noetherian and their quotient is Noetherian

That is to say, M is a R-module and N is a submodule. If N and M/N are both Noetherian, then M is Noetherian. I'm not exactly sure how to begin. I have this so far: Since N is Noetherian it ...
0
votes
2answers
55 views

Subgroups of $\mathbb Z^5$

Is there a nice way to see that any subgroup of $\mathbb Z \times \mathbb Z \times \mathbb Z \times \mathbb Z \times \mathbb Z$ can have at most 5 generators? I know that $\mathbb Z$ is Noetherian, so ...
1
vote
1answer
23 views

Ideal proof in a ring R

Problem Statement: Let I be an ideal in a ring R. Prove that K is an ideal, where $ K $ = { $a\in R$ | $ (\forall r\in R)(ra\in I) $} What exactly am I supposed to show here? I know I need to show ...
1
vote
1answer
64 views

Let $I$ and $J$ be ideals in $R$. Is the set $K= \{ ab \ | \ a\in I, b\in J \}$ an ideal in R? [duplicate]

I've just assumed that this is false, since the problem statements says to compare it to a previous problem where $\{ a+b \ | \ a\in I, b\in J \}$ is ideal. However, by trial and error I can't find ...
2
votes
2answers
55 views

Is there a ring which satisfies $xy=1$ and $yx\neq 1$ [duplicate]

I checked a lot of examples of non-commutative rings that came to my mind, but they weren't helpful. In particularly it's not the case for ring of matrices because of the multiplicity of the ...
0
votes
1answer
188 views

radical membership and ideal membership [closed]

Consider the ideal $I=(x^3y-x^2y^2,x^3z+z^2yx,x^2-xz)\subset \Bbb Q[x,y,z].$ Is $x\in I?$ Is $x\in \sqrt I?$ I'm assuming a question like this is quite simple and that there is just a method, if ...