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Questions tagged [ring-theory]

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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Finding generator for the ideal generated by $a$ and $b$ in an euclidean domain

Let $D$ be a euclidean domain and $a, b \in D$. Show that $M = \{xa + yb \ \mid \ x, y \in D\} $ is an ideal of $D$. Find $d \in D$ such that $M = \langle d \rangle$ and prove your claim. My effort: ...
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1answer
48 views

Is $\mathbb{Z}[x]$ an integral domain? If so, why?

I'm trying to solve a larger problem about maximal and prime ideal, and knowing if $\mathbb{Z}[x]$ is an integral domain would really help me
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2answers
42 views

For prime quadratic integer $\pi$, $x \equiv 1$ (mod $\pi$), Show $x^2 \equiv1$ (mod $\pi^2$) and $x^3 \equiv 1$ (mod $\pi^3$) is not always true.

I was working through one of the problems given to me on my problem set for a number theory class and I would like some help in an attempt to learn. Could someone help me with the following question? ...
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0answers
9 views

left and right regularity in relation to mono and epi?

Consider a category $C$. Suppose $f:a\to b$ is monomorphism. Then for any $g,h:c\to a$ s.t. $fg=fh\implies g=h$. Similarly, one dualizes the statement to obtain the statement for epimorphism. $\...
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3answers
29 views

Prove The Derivative Rules in the Ring of Polynomials

Let R be a commutative ring with unity element 1. Let $f(x)\in R[x]$ and define its derivative as $f'(x)=r_1 +2(r_2)x+...+n(r_n)x^{n-1}$. Prove that $(f+g)'(x)=f'(x)+g'(x)$ and that $(fg)'(x)=f'(x)g(x)...
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0answers
19 views

Compatibility with multiplication of a cyclic order on a ring

Considering a linear order on the additive group of a ring is compatible with multiplication if: $a < b \implies ax < bx$ and $xa < xb$ for any positive $x$, we could define compatibility ...
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1answer
23 views

Showing that $\varphi:\mathbb{Z}/n\mathbb{Z}\rightarrow \mathbb{Z}/m\mathbb{Z}$ is a well-defined surjective ring homomorphism

I have to show that $$\varphi:\mathbb{Z}/n\mathbb{Z}\rightarrow \mathbb{Z}/m\mathbb{Z},a+n\mathbb{Z}\mapsto a+m\mathbb{Z}$$ is a well-defined and surjective ring homomorphism for $m|n$. My idea was ...
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2answers
35 views

Is $R^\infty$ a ring?

Let $(R,+,\cdot,0,1)$ be a ring, and consider the set $$R^\infty=\left\{\{a_n\}_{n=1}^\infty:a_k\in R\text{ for all } k \in \mathbb{Z}_{>0}\right\}$$ with operations $\oplus$ and $\odot$ on $R^\...
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1answer
39 views

Is a finitely generated module over the field of fractions is also finitely generated over the original integral domain?

Let $R$ be an integral domain and $F$ its field of fractions. Let $M$ be a finitely generated $F$-module. Question: Is $M$ also a finitely generated $R$-module? I know that $M$ is an $R$-module ...
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2answers
56 views

$\sqrt{180\mathbb Z} = 30\mathbb Z$, $(180\mathbb Z:700\mathbb Z)= 9\mathbb Z$

(First time studying rings, and I need some help on this example about radical ideal and fraction ideal) Let $\sqrt{I}$ be the radical ideal on the commutative ring $R$, defined as $\sqrt{I}=\{r\in R:...
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0answers
31 views

Why is the normalisation of a singular curve never flat?

In Vakil's notes on Algebraic Geometry, he states in exercise 24.4.H that the normalisation of a singular curve is never flat and he claims that this is a simple consequence of the fact that every ...
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1answer
24 views

reference for theorem commutative ring R is semi simple iff direct product of fields

Can I get any reference(book or journal paper) for the theorem, Let R be a commutative ring, R is semisimple if and only if it is isomorphic to a direct product of a finite number of fields.
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48 views

Is the ideal $I=(X+Y,X-Y)$ in the polynomial ring $\mathbb{C}[X,Y]$ a prime ideal?

Is the ideal $I=(X+Y,X-Y)$ in the polynomial ring $\mathbb{C}[X,Y]$ a prime ideal? Justify your answer. I am a bit hesitant about asking this here. The question is not "How to Solve This Problem". ...
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0answers
37 views

Ring homomorphisms from $\mathbb{R}$ to another unital ring $S$.

We know that there is only one non-trivial ring homomorphism from $\mathbb{Z}$ or $\mathbb{Q}$ to another unital ring $S$. What’s more,when we consider the automorphism of $\mathbb{R}$,it is unique ...
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0answers
31 views

Annihilator and maximal ideal in finite ring

I have this proposition Let $R$ be a finite commutative ring with unity. If $M$ is a maximal ideal in $R$ then $\exists m\in M: M=Ann(m)$ I do not know how to give this $m$ and why the ...
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15 views

Prove that the following conditions are equivalent for a $_RP$ projective module

Let $_RP$ be a projective module, then: End($_RP$) is semiperfect P is semiperfect and finitely generated are quivalent. I have to prove this, but I think I'm not understanding the idea behind. ...
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0answers
29 views

$f$ irreducible over $\mathbb{Z}_{p}$ implies $f$ is irreducible over $\mathbb{Z}$..Why $p$ has to be prime here?

$f$ irreducible over $\mathbb{Z}_{p}$ implies $f$ is irreducible over $\mathbb{Z}$ I think $p$ is supposed to be a prime for the only following reason. I am explaining the reason by taking $p =10$. ...
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3answers
79 views

Showing that $\sqrt{5} \in \mathbb{Q}(\sqrt[p]{2} + \sqrt{5})$

I am attempting to show that $\sqrt{5} \in \mathbb{Q}(\sqrt[p]{2} + \sqrt{5})$, where $p > 2$ is prime. I have already shown that $[\mathbb{Q}(\sqrt[p]{2}, \sqrt{5}) : \mathbb{Q}] = 2p$. If needs ...
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2answers
25 views

Image of an ideal under a surjective ring homomorphism is an ideal

Let $\phi: R \longrightarrow R'$ be a surjective ring homomorphism and $I$ an ideal in $R$. Show that $\phi(I) = \{ \phi (r) : r \in I \}$ is an ideal in $R'$. So I asked this question a couple ...
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2answers
28 views

Describe the set of quadratic integers α in Q[sqrt−3] for which α ̄ and α are associates.

I was working through some textbook problems for my Number Theory class and needed some help with the following question: Describe the set of quadratic integers α in Q[sqrt−3] for which α ̄ and α are ...
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2answers
26 views

Isomorphism between C and another ring

Let the operations of addition and multiplication on the set $K = {at+bu : a,b ∈ R}$, where $t$ and $u$ are formal symbols, be defined as follows: $(at+bu)+(ct+du) = (a+c)t+(b+d)u$, $(at+bu)·(ct+du) ...
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1answer
28 views

Two questions on the ring S := {a+bs : a,b ∈ R}

Let $s$ be a formal symbol. Define addition and multiplication operations on the set $S := {a+bs : a,b ∈ R}$ (with curly brackets) by the rules $(a+bs)+(c+ds) := (a+c)+(b+d)s$, $(a+bs)(c+ds) := (ac+...
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1answer
26 views

Number of ring homomorphism from $\Bbb Z[x,y]$ to $\Bbb F_2[x]/(x^3+x^2+x+1)$

Find the number of ring homomorphism from $\Bbb Z[x,y]$ to $\Bbb F_2[x]/(x^3+x^2+x+1)$. Now I have observed that $(x^3+x^2+x+1)=(x+1)^3$ in $\Bbb F_2[x]$. Then $\Bbb F_2[x]/(x^3+x^2+x+1)=\Bbb F_2[x]/(...
2
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1answer
69 views

Irreducibility of $t ^ { 4 } + t - 1$ over $\mathbb{F}_{25}$

Problem: Let $f ( t ) = t ^ { 4 } + t - 1 \in \mathbb { F } _ { 5 } [ t ]$, show that $f(t)$ has no roots in $\mathbb{F}_{25}$. I tried the following: suppose $f(t)$ has a root in $\mathbb{F}_{25}$, ...
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0answers
37 views

Let $K$ be a ring. Then $K[X_1,X_2] \cong K[X_1][X_2]$

This is a lemma from textbook Analysis I by Amann/Escher. I present my attempt below. Does it look fine or contain gaps/errors? Thank you for your verification! Let $K$ be a ring. Then $K[X_1,X_2] \...
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0answers
39 views

Construction of Roots of Polynomials [on hold]

I'm wondering if I can construct a root of the following three polynomials: $x^2-7x-13$ $x^8-16$ $x^4+x^3-12x^2+7x-1$ I think I can because the field extension of a root and Q is always divisible by ...
3
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0answers
98 views

show $\mathbb{C} [x,y]/(xy-1)$ is a principal ideal domain

I've got to show that $A:=\mathbb{C} [x,y]/(xy-1)$ is a principal ideal domain I know that $A$ is isomorphic to $\mathbb{C}[t,t^{-1}]$ and that this a subfield of $\mathbb{C}[t]$ which is a PID. So ...
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0answers
8 views

Computing a quotient ring. [duplicate]

I wonder how to compute for some $m,n > 0$ the ring $$ (\mathbb{Z}/m\mathbb{Z})/(\overline{n}). $$ I believe it should be the case that $$ (\mathbb{Z}/m\mathbb{Z})/(\overline{n}) \cong \mathbb{Z}/...
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20 views

The class of left serial rings is closed under extensions

A class $S$ is closed under extension if given an ideal $I \subseteq R$ such that $I\in S$ and $R/I\in S$, then $R\in S$. A ring $R$ is left serial if it is a direct sum of left uniserial rings. ...
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0answers
23 views

Suggestion for seminar about rings of continuous functions [on hold]

I have to do a seminar about the rings of continuous functions, it will be a part of a course in topology. The main topic of my seminar will be the functor from the topological space and the ...
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0answers
41 views

Embedding a polynomial ring into $\mathbb{Z}_{n^s}$

My setting is the following: $n$ is a product of two big primes (RSA-like), and I am given a $R = \mathbb{Z}/n^s\mathbb{Z}$ as a space to work with. I would like to represent elements of $R$ as ...
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44 views

An example of an argument using generic points to prove a “closed” condition

Every irreducible affine scheme $\mathrm{Spec}(R)$ contains a generic point, namely $\eta:=\mathrm{Nil}(R)$. If $R$ is a domain then $\eta=(0)$. This is a point which is Zariski dense in $\mathrm{Spec}...
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1answer
26 views

Is every point in the spectrum of a ring $R$ closed?

I am just getting started on spectrums of rings. I see how it is natural to augment the set of prime ideals with the Zariski topology, but from my poor intuition on the topic I don't see how any of ...
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1answer
42 views

Is there a ring whose total ring of fractions is not a field?

I am trying to come up with an example of a ring whose total quotient ring is not a field. I know that if $R$ is a domain, then every total quotient ring has to be a field, however in the general case ...
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0answers
25 views

What's a finite non-commutative ring? [duplicate]

Please give me some examples, this is an exam question of a 1st year math module.
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3answers
34 views

Show that $x^2 +1$ is irreducible in $\mathbb{R}[x]$, but it has roots in $\mathbb{R}[x]\space/\space(x^2 +1) \cong \mathbb{C}$ [duplicate]

So I know that for something to be irreducible, then it cannot be written as the product of non-constant polynomials of smaller degree, but I don't know how to show that the factors don't exist is the ...
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0answers
36 views

Given $\phi \in \mathrm{End}(M)$, when does $\phi$ injective imply $\phi^*$ surjective and $\phi^*$ injective imply $\phi$ surjective?

Let $M$ be a module over a commutative ring (with unity) $R$. Let $\phi : M \to M$ be an $R$-module homomorphism. Then we have a dual map $\phi^* : M^* \to M^*$ given by $\phi^*(f)=f\circ \phi, \...
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1answer
33 views

The generating set of ideals in regular local ring

Let $R$ be a regular local ring. It is well-known that $R$ is a Cohen-Macaulay ring. Hence $grade(I,R)=ht(I)$ if $I$ is an ideal. If $I$ is a proper ideal, suppose $ht(I)=d$, is there exists $x_1,.....
1
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1answer
29 views

Equivalence for rings with localization property

I feel like I need a hint for the following exercise: Let $R$ be some commutative unitary ring. If $M$ is a $R$-module, let $M[f^{-1}]$ denote the localization of $M$ with respect to the set $\{ f^...
1
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1answer
28 views

Non-unital algebra homomorphism with strange property

Let $A$ be a unital not-necessarily commutative algebra, defined over $\mathbb{R}$ or $\mathbb{C}$. Take some $\alpha$ a non-unital algebra automorphism of $A$. Is it possible to find an example for $...
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0answers
64 views

Given a split exact sequence $0 \to N \to M \to M \to 0$, when can we say $N=0$?

Let $M$ be a module over a commutative ring $R$. Let $N$ be a submodule of $M$ such that there is a split exact sequence $0 \to N \to M \to M \to 0$. So, in particular, $M \cong M \oplus N$. Under ...
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1answer
35 views

Hopfian modules and equivalence of categories of modules

For a ring with unity (not necessarily commutative) $R$, let $R$-$Mod$ denote the category of left $R$-modules. Let $R,S$ be two rings with unity and $T: R$-Mod $\to S$-Mod be an equivalence of ...
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1answer
36 views

In a ring, does $a^2=0$ imply $a=0$?

Let $R$ be a ring and $a$ be an element of $R$: $a^2 = 0$. Must it be true that $a = 0$? (Assuming $0$ is the additive identity of $R$)
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1answer
38 views

If $\phi:R\rightarrow R'$ is a surjective ring homomorphism and I is an ideal in R… continued below [duplicate]

then $\phi(I)=[s'\in R'|s=\phi(s)\space \forall\space s\in I]$ is an ideal in R' So I know that for something to be an ideal, it needs to be closed under subtraction and it must absorb products. I ...
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2answers
31 views

Prove: If $\phi:R\rightarrow R'$ is a ring homomorphism, then $image(\phi)$ is a subring of $R'$

Do I have to use the regular axioms for proving something is a subring? i.e. closed under subtraction and multiplication. If so, can I say $$im(r-s)=r-s\in R$$ $$im(rs)=rs\in R$$ Therefore $im(\phi)$ ...
1
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1answer
31 views

Is a finitely generated subring of a Noetherian ring also Noetherian?

Is a finitely generated subring of a Noetherian ring $R$ also Noetherian? Remark: In fact I'm interested in the case $R=\mathbb C[x_1,...,x_n]$.
1
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1answer
26 views

A sufficient condition for a unitary ring to be a local ring

Theorem. Let $R$ be a unitary ring such that $R$ is a subring of a division ring $D$. If for all $d(\ne 0)\in D$ either $d\in R$ or $d^{-1}\in R$ then $R$ is a local ring. My Proof. It suffices ...
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2answers
40 views

Reduced and integral rings

Let $R$ be a commutative ring with unit. Are the following true? If $\operatorname{Spec}(R)$ is irreducible i.e, cannot be written as union of two proper closed subsets, and $R$ is reduced, then $R$ ...
0
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1answer
20 views

Prove: If $\mathit R$ is a commutative ring with unity and $\mathit I=(x)\subseteq R[x]$, then $R[x] / (x)\cong R$ [duplicate]

I know that to show a ring is isomorphic to another ring, I have to find a bijective ring homomorphism between the two rings. Or I could use the F.H.T. but I would also need a function to make that ...
0
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0answers
18 views

Transitivity of integral extensions and prime ideals

The situation is as follows: We have $K$ a field $K[a_1, ..., a_n] \subseteq R$ finite ring extension $R \subset R'$ integral ring extension of integral domains Since $R$ is finite over $K[a_1, .....