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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1
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1answer
22 views

Finite dimensional division algebra over C

Another abstract algebra question from my university days that has me stumped at where to start! I know what a division ring is and I think I understand what a division algebra over $\mathbb C$ is. ...
2
votes
1answer
27 views

In general, when does a ring have a division algorithm?

I'm working through Herstein's "Abstract Algebra" text, and am currently working through section 5. Theorem 4.5.5 introduces the division algorithm for polynomial rings over fields, which states: ...
3
votes
2answers
15 views

$il+M=1+M \implies il =1$ or $il=1+m, m\in M$, hence $I=R$

If we have $R/M$ is a field and $M,I$ are ideals of $R$ such that $M\subseteq I \subseteq R$. If we take $i\in I, i\not\in M$ we have $i+M \ne 0+M$. Since $R/M$ is a field, we have that $i+M$ is ...
0
votes
0answers
25 views

About two polynomials $f,g$ such that $f=\pm g$

Let $R$ be an infinite commutative ring with unit and with characteristic zero. Assume that $f,g\in R[x_1,...,x_n] $ are nonzero and such that $f(x_1,...,x_n)=s(x_1,...,x_n) g(x_1,...,x_n)$, where ...
4
votes
1answer
33 views

How do I use homomorphism theorem to show the assertion?

Show that $\mathbb Z[x]/\langle x^2-3,2x+4 \rangle$ is isomorphic to $\mathbb Z_2[\sqrt 3]$. I tried to use first homomorphism theorem, but not able to get that how should I approach.
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0answers
19 views

How do I show this assertion? [duplicate]

Show that the ideal generated by $x^2-y$ is a prime ideal in $C[x,y]$. It would be sufficient if we show that $C[x,y]/<x^2-y>$ is an Integral Domain. Or is there any other way of showing the ...
2
votes
0answers
34 views

Nilradical of a polynomial ring

I am asked to compute the $nilrad(\mathbb{C}[X])$ and the reduction $\mathbb{C}[X]_{red}$. $\textbf{DEFINITION:}$ An element $a \in R$ is nilpotent if $a^n = 0$ for some positive integer $n$. ...
4
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0answers
43 views

Vakil's definition of smoothness — what happens at non-closed points?

The following is definition 12.2.6 in Vakil's notes. A $k$-scheme is $k$-smooth of dimension $d$, or smooth of dimension $d$ over $k$, if it is pure dimension $d$, and there exists a cover by ...
1
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1answer
25 views

Let $R$ be a ring. Let $I\lhd R$ and fix $n\in I$ if $n$ is the unit of $R$. Show $R=I$

Let $R$ be a ring. Let $I\lhd R$ (that is $I$ is an ideal of the ring) and fix $n\in I$ if $n$ is the unit of $R$. Show $R=I$. Here is my attempt at an answer: We aim to show $I \subseteq R$ and $R ...
2
votes
1answer
37 views

Constructing a quotient ring in GAP using structure constants [on hold]

I need to construct the following ring in GAP: $$Z_4(u) / \langle u^2-2u=0 \rangle. $$ This is what I tried and it didn't work: ...
1
vote
1answer
27 views

Let $d \in \mathbb{Z}$, $d > 1$. Determine all the ideals of $\mathbb{Z}/d\mathbb{Z}$ which are prime or maximal

Let $d \in \mathbb{Z}$, $d > 1$. Determine all the ideals of $\mathbb{Z}/d\mathbb{Z}$ which are prime or maximal I know that $m\mathbb{Z}/md\mathbb{Z} \cong \mathbb{Z}/d\mathbb{Z}$ as rings ...
0
votes
1answer
20 views

Constructing a quotient ring of multivatiate polynomial ring in GAP

I need to construct the following ring in GAP: $$F_2(u_1,u_2) / \langle u_1^2=u_2^2=0,u_1u_2=u_2u_1 \rangle $$. This is what I tried and it didn't work: ...
4
votes
1answer
29 views

Is it true if $R = mZ/mdZ$ is isomorphic to $Z/dZ$, then it must have a unit element?

Is it true if $R = mZ/mdZ$ is isomorphic to $Z/dZ$, then it must have a unit element? This is a question I ask myself, but I'm not certain of this answer. Is anyone could explain to why this is (or ...
0
votes
1answer
32 views

What are the ideals of the ring $\mathbb{Z}[x]/(2,x^3+1)$? [on hold]

What are the ideals of the ring $\mathbb{Z}[x]/(2,x^3+1)$? I'm stuck at how to determine what ring this ring is isomorphic to?
1
vote
0answers
20 views

Prime and Maximal Ideals of $\mathbb{Z}[x]$ [duplicate]

Consider $R=\mathbb{Z}[x]$. Also let $p$ be a prime. Then we want to find all the prime and maximal ideals of $\mathbb{Z}[x]$. The prime ideals are $(0), (p), (x)$ and $(ap + bx)$. Then we see that ...
-1
votes
1answer
37 views

P.I.D. and a nontrivial ideal, Quotient ring has finitely many ideals [on hold]

A ring $R$ is a P.I.D. Let $I$ be a nontrivial ideal in $R$. Prove that $R/I$ has finitely many ideals.
2
votes
0answers
40 views

defining gcd on rings

I see that in most textbooks they say let $R$ be an integral domain and start defining the greatest common divisor. My question is, can gcd's be defined on just commutative rings without an identity?
1
vote
1answer
46 views

A subset of a polynomial ring and its ideal. [duplicate]

Let $P=K[x_1,\dots,x_n]$ be a polynomial ring over a field $K$ and $I = (f)$ be a principal ideal in $P$ generated by $f \in P - \{0 \}$. Moreover let $L \subset \{x_1, \dots, x_n \}$ and $\hat{P} ...
1
vote
0answers
29 views

$K$-affine $Hom$ functor relating to Polynomial Maps (Exercise in _Algebra: Chapter 0_ by Aluffi)

I'm currently learning some category theory and algebraic geometry, the basics at least. In Algebra: Chapter 0 by Aluffi, there is an exercise describing the relationship between morphisms of affine ...
3
votes
2answers
35 views

If $|F| = 8$ and $K$ is a subfield, show that $K = F$ or $K = \{0, 1\}$.

Let $K$ be a subring of a field $F$. If $|F| = 8$ and $K$ is a subfield, show that $K = F$ or $K = \{0, 1\}$. [Hint: Lagrange]. Lagrange's Theorem: If $H$ is a subgroup of a finite group $G$ I Then ...
1
vote
2answers
30 views

Show that a ring $R$ is a division ring if and only if, for each nonzero $a\in R$, there is a unique element $b\in R$ such that $aba = a$. [duplicate]

Show that a ring $R$ is a division ring if and only if, for each nonzero $a\in R$, there is a unique element $b\in R$ such that $aba = a$. $\Rightarrow$ Assume $R$ is a division ring. Let ...
1
vote
1answer
33 views

Constructing a ring F_2(u)/<u^2=0> in GAP

I need to construct the following ring in GAP: $$F_2(u) / \langle u^2=0 \rangle =\{ \; a+bu \; | \; a,b \in F_2 \; \}=\{0,1,u,1+u\}$$. I tried using the commands ...
0
votes
2answers
23 views

Cosets of submodule

Suppose we have a ring $R$ and an ideal $J \subseteq R$. Let $M$ be an $R$-module and $N$ be a submodule of $M$. Assume for two elements $m$ and $m'$ of $M$ we have $$m+ JM=m'+JM.$$ Since $N ...
0
votes
1answer
30 views

Questions abaout ring [on hold]

I have a comutative ring (R,+,.) with unity then i have to say which of the following is true: A[X]=A[[X]] A[X] included in A A[X] included in A[[X]] A[[X]] included in A[X] I can't figure it out.
0
votes
1answer
51 views

Show that no ring containing R can contain a root of g(x) = 3x +1

Show that if $R = \mathbb Z_6$ and $g(x) = 3x + 1 ∈ R[x]$, then $R[x]/(g(x)R[x])$ does not contain a root of $g(x)$. More generally, show that no ring containing $R$ can contain a root of $g(x)$. ...
1
vote
2answers
49 views

Ring of matrices has no nontrivial ideals [duplicate]

It is a theorem that a commutative ring is a field if and only if it has no nontrivial ideals. Clearly this does not hold in the noncommutative case. I am trying to show for instance that the ring of ...
4
votes
2answers
52 views

For which rings does a polynomial in $R$ have finitely many roots?

Which infinite rings satisfy the following Every non-zero polynomial in $R[X]$ has only finitely many roots ? Are there such rings which are not integral domains ?
0
votes
1answer
23 views

A $R$-module over a ring $R$ with identity is free iff it is a direct sum of copies of $R_R$, where $R_R$ is $R$ considered as a module over itself

Let $R$ be a ring with identity and $M$ be an $R$-right module. Then $M$ is called free over $X \subseteq M$ if for every module $N$ with mapping $\alpha : X \to N$ we can extend it uniquely to a ...
0
votes
1answer
29 views

Show that $m\mathbb{Z}/md\mathbb{Z} \cong \mathbb{Z}/d\mathbb{Z}$ if and only if $(d,m)=1$

Show that $m\mathbb{Z}/md\mathbb{Z} \cong \mathbb{Z}/d\mathbb{Z}$ as rings without identity if and only if $(d,m)=1$. I know that if $\phi : A \to B$ is a epimorphism ring and $A$ is a unit ...
1
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0answers
64 views

Solving a polynomial for cyclic roots

For any complex value of c, the following polynomial has 6 complex root values of p: $$1+c+2 c^2+c^3+p+2 c p+c^2 p+p^2+3 c p^2+3 c^2 p^2+p^3+2 c p^3+p^4+3 c p^4+p^5+p^6$$ For a general 6th order ...
0
votes
0answers
10 views

Representation of left ideals of the matrix ring [duplicate]

We know that $J$ is an ideal of the full matrix ring $S=M_n(R)$ if and only if $J$ is the ring of all $n\times n$ matrices over $I$ for some ideal $I$ of the ring $R$ with identity. Now, my question ...
2
votes
1answer
42 views

Subrings of the product of rings of algebraic integers

Let $K,L$ be a number fields and $\mathcal{O}_K,\mathcal{O}_L$ their rings of algebraic integers, and $n_K=[K:\mathbb Q]$, $n_L=[L:\mathbb Q]$. Suppose that $R$ is a subring of ...
5
votes
0answers
58 views

Prove that $u$ is a trivial unit

Let $G$ be a finite group. I want to prove that if $u\in \mathbb{Z}G$ be a torsion unit (i.e. for some $n$, $u^n=1$) of integral group ring $\mathbb{Z}G$ such that $u$ normalizes $G$, then $u$ is a ...
3
votes
1answer
32 views

Equivalent properties of Von Neumann regular rings

Let $M$ be a module over a ring $A$ and $R=Hom_A(M,M)$ its endomorphism ring (with respect to the composition). I need to show these following conditions are equivalent: $\alpha = \alpha \beta ...
0
votes
2answers
65 views

Show that $\phi(p^e)=p^e-p^{e-1}$

In an exercise I was asked to show that if $R$ is a ring with relatively prime ideals $I_1,I_2$ then $R/I \cong R/I_1 \oplus R/I_2$ where $I=I_1 \cap I_2$ and $\oplus$ is the direct sum. A follow on ...
1
vote
1answer
33 views

If I is an irreducible ideal, and P is a prime ideal, is (I+P)/P irreducible?

Let $A$ be a commutative ring with unit, and $P$ a prime ideal. My question is: If $I$ is an irreducible ideal in $A$, is $(I+P)/P$ irreducible in $A/P$? If not, can you show a counterexample? ...
0
votes
1answer
42 views

Noncommutative rings and the evaluation homomorphism

Recall the evaluation homomorphism of a ring. For example, if $\{R[x]|{p(x)=a_0+a_1x+a_2x^2...}$} is the ring of polynomials with real coefficients then we can evaluate with respect to $c$ by letting ...
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0answers
35 views

Matrices representing a map between free modules of infinite rank and Fitting's Lemma (Eisenbud)

p.497 of Commutative Algebra with a View Toward Algebraic Geometry, Eisenbud: If $\phi: F \rightarrow G$ is a map of free modules, then $I_j\phi$ is the image of the map $$\Lambda^j F ...
1
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3answers
59 views

For which $n \in \mathbb{N}$ does $x^8 + [2]_nx^7+[6]_nx^6-x^2-[2]_nx - [6]_n = [0]_n$ have at least $7$ distinct solutions?

I have to find one $n \in \mathbb{N}$ such that $$x^8 + [2]_nx^7+[6]_nx^6-x^2-[2]_nx - [6]_n = [0]_n$$ has at least $7$ distinct solutions in $\mathbb{Z}_n$ (or, equivalently, $f(x) = x^8 + ...
0
votes
1answer
49 views

Correspondence between prime and maximal ideals [closed]

My professor put the following statement in the lecture notes without proof: Let $R$ be a commutative ring and $I$ an ideal. Then the natural correspondence between ideals containing $I$ and ideals ...
1
vote
1answer
44 views

Intersection of two ideals

Let $A$ be a commutative ring and let $\mathfrak{a}$, $\mathfrak{b}$ be ideals in $A$. I am asked the following question: Show that $\mathfrak{a} \cap \mathfrak{b}$ is the largest ideal of $A$ ...
1
vote
2answers
22 views

Is the set of all $f$ such that $\lim_{x\to1^-}f(x) = 0$ an ideal in the ring of functions from $[0,1]\rightarrow \mathbb{R}$?

Is the set of all $f$ such that $\lim_{x\to1^-}f(x) = 0$ an ideal in the ring of functions from $[0,1]\rightarrow \mathbb{R}$? I'm sure about the closure under addition but not quite clear about if ...
4
votes
0answers
64 views

Generalisation of euclidean domains

Recently I wondered how dependent the definition of euclidean domains is of the co-domain of the norm-function. To be precise: Let's define a semi-euclidean domain as a domain $R$ together with a ...
1
vote
2answers
45 views

An irreducible polynomial in a subfield of $\mathbb{C}$ has no multiple roots in $\mathbb{C}$

Let $K\subset \mathbb{C}$ be a subfield and $f\in K[t]$ an irreducible polynomial. Show that $f$ has no multiple roots in $\mathbb{C}$. If I understand this question correctly, I must show that ...
0
votes
1answer
29 views

$f$ is divisible by a square of non-constant polynomial iff $f,f'$ are not relatively prime

Let $R$ be a commutative ring and $f=a_0+ \cdots +a_nt^n \in R[t]$. Define $f':=a_1+2a_2t+ \cdots + na_{n-1}t^{n-1}$. Show that $f$ is divisible by a square of non-constant polynomial if and only ...
0
votes
2answers
21 views

Union Over a Totally Ordered Set of Ideals is an Ideal

I am trying to understand the proof of a theorem which uses Zorn's lemma. I understand quite well all parts of the proof except for one point: Let $R$ be a ring and define $K\doteq \{I\subseteq R ...
-1
votes
1answer
30 views

Does localization of a Noetherian ring always give a local ring? [closed]

I have a local ring $A$ and suppose I localized this ring at prime $P$. Is the localized ring $A_P$ a local ring? I was wondering if it requires additional properties on $A$. Thank you very much!
-2
votes
1answer
59 views

Property of a Noetherian ring: How come $P \setminus P^2$ is non-empty? ($P$ is a prime ideal) [closed]

Let $A$ be a Noetherian ring, and let $P$ be a prime ideal. How come we know that $P \setminus P^2$ is non-empty? Thank you!
1
vote
1answer
30 views

Determining all the homomorphisms $\mathbb{Z} \to R$, where R is an integral domain.

I think I have this question figured out almost completely, but I'm a little worried about using a certain notation. Suppose $\mathbb{Z} \stackrel{\phi}{\longrightarrow} R$ is a ring homomorphism. ...
0
votes
1answer
18 views

Computing generators for a finitely generated module

I came across this problem yesterday: Let $R$ be a ring and $M$ an $R-$module. $\varphi:R^n\to M$ is a surjective $R-$module homomorphism if and only if $M$ is finitely generated. Given the set of ...