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
28 views

set theoretic equivalence in quotient ring

If I am given a ring $R$ and a 2-sided ideal $K\subseteq R$, I know that I have a well-defined quotient ring $R/K$. My question is the following: We know that if we have $a,b\in R$, then ...
0
votes
0answers
13 views

About ring principal ideal [duplicate]

Have this homework assigment, able to do it, except last part. Here is exercise: 1.Prove that set $I=\{f(x)\in\mathbb{Z}[x]|f(0) \quad is\quad even\}$ is rings $\mathbb{Z}[x]$ ideal 2.Prove that if ...
0
votes
0answers
19 views

Showing a finite field is a local ring

I am asked to find an integer $n$ such that $\mathbb{Z}_n$ is a local ring and then to prove my answer. Here is my attempt: We know that $Z_p$ is a finite field for prime $p$. Since $\mathbb{Z}_p$ ...
1
vote
1answer
29 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
32 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: ...
7
votes
2answers
176 views

Maximal Ideals in Ring of Continuous Functions

Dummit and Foote, 7.4.33(a): Let $R$ be the ring of all continuous functions $[0,1] \to \mathbb{R}$ and let $M_c$ be the kernel of evaluation at $c \in [0,1]$, i.e. all $f$ such that $f(c) = 0$. ...
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: ...
1
vote
1answer
34 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 ...
2
votes
1answer
40 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: ...
3
votes
2answers
16 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
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0answers
26 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
36 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.
-1
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0answers
20 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
42 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
votes
0answers
45 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
28 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 ...
1
vote
1answer
26 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 ...
4
votes
1answer
30 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
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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
vote
1answer
47 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
votes
1answer
38 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.
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votes
3answers
181 views
+200

Is $\mathbb{Z}[\sqrt{15}]$ a UFD?

Let $R=\mathbb{Z}[\sqrt{15}]=\{a+b\sqrt{15}:a,b\in\mathbb{Z}\}$. How do I show that $(3,\sqrt{15})$ is a maximal ideal but not a principal ideal? How do I show that $(3,\sqrt{15})^2$ is a ...
3
votes
2answers
76 views

In a ring $(A,+, \cdot)$ if $aba = a$ then $bab = b$ and all non zero elements in $A$ are invertible. [on hold]

Let $\left(A,+, \cdot\right)$ be a ring with $1$ that satisfies the following condition: For any nonzero $a\in A$, there exists a unique $b\in A$ such that $aba = a$. Show that $b$ also satisfies ...
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 ...
2
votes
0answers
41 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?
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0answers
46 views

Groebner basis and prime ideals.

Let $I$ be an ideal in a polynomial ring $P = K[x,y_1,\dots,y_n]$ and assume that $I \cap K[x]\neq (0)$. Let $>$ be an elimination ordering for $\{y_1, \dots, y_n\}$ and $G$ is a Groebner basis for ...
7
votes
4answers
640 views

Do properties of algebraic structures sometimes not carry over when their direct products are taken?

I recently had a homework problem asking to prove that the direct product of rings (or rings with identity) are still rings (with identity), and it seemed really silly to go through all the steps in ...
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 ...
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 ?
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
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0answers
30 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 ...
1
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0answers
72 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 ...
4
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0answers
65 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 ...
10
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3answers
1k views

Ideal in the product of two rings

$R$ and $S$ are two rings. Let $J$ be an ideal in $R\times S$. Then there are $I_{1}$, ideal of $R$, and $I_{2}$, ideal of $S$ such that $J=I_{1}\times I_{2}$. For me is obvious why $\left\{ r\in ...
0
votes
1answer
31 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
52 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
50 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 ...
0
votes
1answer
24 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 ...
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0answers
29 views

twisting rings for FFT

I am implementing an FFT described by Daniel Bernstein in http://cr.yp.to/papers.html#multapps on page 332 (8 in pdf) he states the following: One can multiply in $R[x]/(x^{2n} +1)$ with $(34/3)n ...
0
votes
1answer
30 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 ...
6
votes
2answers
89 views
+50

Is there a finite ring whose rank is smaller than the rank of its group and its monoid?

Consider a finite ring $(R, +, \times)$ comprising a finite additive abelian group $(R, +)$, a finite multiplicative monoid $(R, \times)$, and a distributivity rule relating the two. Let the rank of ...
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
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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 ...
2
votes
2answers
203 views

Example of an integral domain that is not integrally closed and having some localization which is also not integrally closed

Can anyone show an example of integral domain that is not integrally closed and also has one of its localization with respect to a maximal ideal not integrally closed?
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
73 views

There exists a zero dimensional ideal I such that $\dim (R/I) - |V(I)| \geq \alpha > 0$

If $I \subset K[x_1,\dots,x_n]$ is a zero dimensional ideal and $$V(I) = \{ (\alpha_1,\dots,\alpha_n) \in K^n: f((\alpha_1,\dots,\alpha_n)) = 0\ \forall f\in I\}$$ (the variety). I know that ...