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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3
votes
3answers
49 views

Does there exist a unital ring whose underlying abelian group is $\mathbb{Q}^*$?

Let $\mathbb{Q}^*$ be the group of units of the rational numbers. Does there exist a unital ring whose underlying additive group is $\mathbb{Q}^*$? I don't really have a gut feeling yea or nea. ...
0
votes
1answer
27 views

judge if nilradical equals jacobson radical

judge if nilradical equals jacobson radical 1)a noetherian ring that is not a artin ring. 2)a local integral domain that is not a field. 3)a integral domain with only finite number of ...
0
votes
1answer
30 views

Example of non noetherian ring and noetherian $\Bbb Z$-module

a non Noetherian ring that is a Noetherian $\Bbb Z$-module a Noetherian ring that is a non Noetherian $\Bbb Z$-module I have no idea in 1, and I'm not sure if $\mathbf{Q}$ is right for 2? ...
-2
votes
1answer
46 views

Why field of fractions of $k[x_1,x_2,…]$ is Noetherian? [on hold]

the classical counterexample of a subring of a noetherian rings that is not noetherian is $k[x_1,x_2,...]$, which is not noetherian, but the field of fractions of $k[x_1,x_2,...]$ is, can anyone ...
0
votes
1answer
11 views

$R$ is the ring $\mathbb{Z}[\sqrt{-k}]$. In $R$, if $3\mid (a+b\sqrt{-k})$, then $3\mid a$ and $3\mid b$ in $\mathbb{Z}$

$R$ is the ring $\mathbb{Z}[\sqrt{-k}]$ where $k\ge 5$ and $k\equiv2 \pmod{3}$. I would like to prove that in $R$, if $3\mid(a+b\sqrt{-k})$, then $3\mid a$ and $3\mid b$ in $\mathbb{Z}$. I have ...
0
votes
2answers
44 views

every ideal is contained in a maximal ideal

The statement is: In a commutative ring with 1, every proper ideal is contained in a maximal ideal. and we prove it using Zorn's lemma, that is, $I$ is an ideal, $P=\{I\subset A\mid A\text{ is ...
0
votes
2answers
38 views

Example of ideals such that $I^n=0$ but $I^{n-1}\not= 0$

Let $R$ be a ring. For each $n>0$ I want to find an ideal $I$ of $R$ such that $I^n=0$ but $I^{n-1}\not= 0$. Clearly this won't work for $R=\Bbb{Z}$ or $\Bbb{Z}/n\Bbb{Z}$. And I ran out of ...
2
votes
1answer
40 views

Are the two ways of creating an $S^{-1}A$ algebra equivalent?

Let $f:A\to B$ be a ring homomorphism and $S$ be a multiplicative set, define $S^{-1}B$ to be $B\times S$ with equivalence relation $(b,s)\sim(b',s')$ iff $\exists t\in S$ such that $t(sb'-s'b) = 0$. ...
1
vote
2answers
41 views

MCS meet all prime ideals

let A be a commutative ring, is there any multiplicatively closed subset S (not containning 0), s.t. every prime ideal in A intercept S is not empty? My thinking is that there is 1-1 ...
2
votes
1answer
66 views

Why $ \mathbb{Z}[x]$ is not Principal Ideal Domain [duplicate]

$ \mathbf{Z}[x]$ is not PID. we know $\mathbb Z$ is a Unique Factorization Domain, so $\mathbb Z[x]$ is UFD, but why isn't it PID (since I think $\mathbb Z$ is PID)?
1
vote
2answers
46 views

Minimal subring of complex numbers

Let $\alpha$ be a root of $X^3+X^2-2X+8$. My question is if $\mathbb Z\left[\alpha,\frac{\alpha+\alpha^2}{2}\right]=\{a+b\alpha +c\frac{\alpha+\alpha^2}{2}:a,b,c\in\mathbb Z\}$? Thank you all.
5
votes
2answers
154 views

Where does the proof for commutative rings break down in the non-commutative ring when showing only two ideals implies the ring is a field?

We know in a commutative ring, if the only ideals are trivial and the whole ring, then the ring is a field, which is proved by every ideal is contained in a maximal ideal, which is proved by Zorn's ...
0
votes
1answer
47 views

If R is an integral domain disprove the RxR is an integral domain? [duplicate]

I am trying to prove that given R (an integral domain) it is not true that then RxR is an integral domain: We know that for the ring Zp for any prime p, Zp is an integral domain because it is a ring ...
-4
votes
0answers
51 views

Give an example of a field K, an extension field F, a subring R of F containing K where R is not a field? [on hold]

Give an example of a field K, an extension field F, a subring R of F containing K where R is not a field? I had $\mathbb{C}$ as a field, $\mathbb{C}(x)$ as a field extension, and $\mathbb{C}[x]$ ...
1
vote
3answers
34 views

Is is true that $R[x,y]/(x-y^2) \cong R[y]$?

I have a ring $R$ and I want to prove (or disprove) that $R[x,y]/(x-y^2) \cong R[y]$. My idea is to define a ring homomorphism $\phi$ such that $\phi$ is the identity on $R$ and such that $\phi(y) = ...
2
votes
3answers
63 views

Injective Ring Homomorphism

I seem to be having the wrong impression of what $p$ stands for; is $p(x)=x(x+1)(x+2)$ or is it something else? Clarification would be appreciated so that I can complete the lemma below. Consider ...
3
votes
0answers
24 views

To prove ; $pa:=a+a+… p $ times $=0 , \forall a \in R$ and $a^p=a , \forall a \in R$ for some prime $p$ then the ring $R$ is commutative

If in a ring $R$ , $\exists $ prime $p$ such that $pa:=a+a+... p $ times $=0 , \forall a \in R$ and $a^p=a , \forall a \in R$ , then how to prove that $R$ is commutative ? I would not want to use ...
0
votes
2answers
57 views

Solution verification: proving that $2$ is not prime in $\mathbb{Z}[\sqrt{-3}]$

I just took my final exam for abstract algebra and have this problem stuck in my head. Prove that $2$ is irreducible in $\mathbb{Z}[\sqrt{-3}]$ but not prime. My Solution: Proving that it is ...
3
votes
3answers
34 views

Confused about a solution: proving every prime ideal is maximal

I'm looking at this solution to this problem: I'm getting thrown off by the special case where $n = 2$. If $n = 2$, why must it be that $x = 1$? All that we then know is that $x^2 = 1$ or that $x = ...
1
vote
2answers
39 views

For any $p,q\in\mathbb{Z}[i]$, $\mathrm{N}(\gcd(p, q))$ must divide $\gcd(\mathrm{N}(p), \mathrm{N}(q))$

I'm studying for my final exam (abstract algebra) and am looking at an example where our professor was trying to compute the GCD of two elements of $\mathbb{Z}[i]$. Rather than directly applying the ...
3
votes
1answer
37 views

Irreducible elements and unique factorization domain

Let $P=\{\frac{a}{3^n} : a \in \mathbb{Z}, n \in \mathbb{N}\}$. a) Which elements are irreducible in $P$: 4, 5, 6, 9, 10, 15? b) Find out, which one of rings: $ P$, $\mathbb{Z}[i\sqrt{5}]$, $P[x]$ ...
1
vote
1answer
25 views

Showing that an epimorphism of an ideal is again an ideal

Let $R, S$ be commutative rings, $f : R \rightarrow S$ an epimorphism, I an ideal of R. Show that $f(I)$ is an ideal of $S$. As far as I understand, I need to show 4 things: 1) $0_s \in f(I)$ ...
0
votes
1answer
67 views

Is ideal prime or maximal? [on hold]

Find, whether or not given ideal of $\mathbb{Z}[x]$ ring is prime or maximal and describe the quotient ring : a) $J_1 = (x-5)$ b) $J_2 = (3, x+5)$. How can I do that?
1
vote
1answer
30 views

For $f, g \in K[t]$, $f \neq g$ implies $f_K \neq g_K$

Consider an infinite field $K$. For $f, g \in K[t]$, show that $f \neq g$ implies $f_K \neq g_K$, where $f_K, g_K: K \rightarrow K$ denote the usual polynomial functions. My attempt: By ...
2
votes
1answer
28 views

Example of finite ring with a non principal ideal

I would like to have an example (or a class of examples) of a finite ring (with unit, not necessarily commutative) that is not a principal ideal ring (if possible an example of a local ring and of ...
0
votes
1answer
25 views

Proving existence for a combination of mappings from a group to a commutative ring

Let $G$ be a group and $R$ be a commutative ring. Let $X$ be the set of all mappings $\phi : G \rightarrow R$ with $\phi(g) \neq 0$ for finitely many $g \in G$. For all $g \in G$ define $$(\phi_1 ...
0
votes
1answer
33 views

Proving relations between (sub-)rings and a group

Let $R \neq 0$ be a commutative ring, $G$ be a finite group, $\#G > 2$. 1) $H$ subgroup of G $\Rightarrow$ monoid ring R[H] is a subring of monoid ring R[G] 2) Let $x := \sum_{g\in ...
-1
votes
2answers
51 views

Trying to understand a proof for the automorphisms of a polynomial ring

Consider the following proof for finding all automorphisms of the ring $\mathbb{Z}[x]$ which I am trying to understand. Source I have two question regarding the proof 1) They set $d = ...
1
vote
3answers
63 views

Is $\mathbb{Q}[x]/(x-1)(x^2+4)\mathbb{Q}[x]$ isomorphic to $\mathbb{Q}\times\mathbb{Q}[i]$?

Problem: Check if $\mathbb{Q}[x]/(x-1)(x^2+4)\mathbb{Q}[x]$ is isomorphic to $\mathbb{Q}\times\mathbb{Q}[i]$. If not, find what is it isomorphic to. My guess: they're isomorphic. My attempt: I ...
1
vote
4answers
84 views

trying to understand what a polynomial ring is

I don' really understand what a polynomial ring is, maybe because the lack of examples. Consider for example the polynomial ring $\mathbb{Z}[x]$. Can you please tell me how this polynomial ring (its ...
0
votes
1answer
37 views

maximal ideal problem [duplicate]

I want to solve this problem, but I have no idea how I can start: If $K$ is a field, $(a_1,...,a_n) \in K^n,$ and $I$ the ideal $I=\langle x_1-a_1,...,x_n-a_n\rangle$, then how can we prove that ...
5
votes
2answers
136 views

Is this particular module flat?

Let $A=k[x^2,xy,y^2]\hookrightarrow B=k[x,y]$, where $k$ is a field. Is $B$ flat over $A$? I am guessing the answer is no. My first thought is, since $B$ is integral over $A$, so it's finitely ...
9
votes
2answers
169 views

Artin Chapter 11, Exercise 9.12, polynomials without common zeroes [on hold]

How do I show that the three polynomials $f_1 = t^2 + x^2 - 2$, $f_2 = tx - 1$, $f_3 = t^3 + 5tx^3 + 1$ generate the unit ideal in $\mathbb{C}[t, x]$? Artin mentions two approaches: by showing that ...
1
vote
2answers
65 views

On a proof that left artinian implies left noetherian

Questions: [Refer to below] Could one elaborate on $\rm\color{#c00}{(a)}$, $\rm\color{#c00}{(b)}$ and $\rm\color{#c00}{(c)}$ ? My thoughts : $\rm\color{#c00}{(a)}$ For $r+J\in R/J$ and ...
1
vote
1answer
36 views

Solution of $ax+xb=c$ in a division ring

The equation $ax+xb=c$ in the quaternions skew field ($a,b,c,x \in \mathbb{H}$) has solution: $$ x= \left(|b|^2+2b_0a +a^2\right)^{-1} \left( ac +c \bar b\right) $$ Where $|b|,b_0,\bar b$ are ...
3
votes
0answers
24 views

Example of finite ring which is not a Bézout ring

A left (or right) Bézout ring is a ring in which any two elements generate a principal left (resp. right) ideal. Assume that we have a finite ring R. Does there exist some classification theorem ...
4
votes
2answers
42 views

Why $P_1\neq P_1P_2$?

Question: If $P_1,P_2$ are distinct prime ideals of an artinian ring, why is it that $P_1\neq P_1P_2$? I know that prime ideals of an artinian ring are maximal, but still, I can't see why ...
0
votes
1answer
45 views

Compute the index of $\mathbf{Z}[\alpha]$ in $\mathbf{Z}[\alpha,\beta,\gamma]$

How do I compute the index of $\mathbf{Z}[\alpha]$ in $\mathbf{Z}[\alpha,\beta,\gamma]$? For example, $\alpha = \sqrt[3]{-19}$ and $\beta = (\alpha^2 - \alpha + 1)/3$ satisfy $(\alpha + 1)\beta = ...
3
votes
1answer
23 views

bilinear maps with respect to noncommutative rings

Consider a noncommutative ring with unity $R$, three left $R$-modules $M,N,P$ and a map $f\colon\;M\times N\to P$ such that: $ f(m+m',n)=f(m,n)+f(m',n)\\ f(m,n+n')=f(m,n)+f(m,n')\\ ...
3
votes
2answers
74 views

The interpretation of ideals of a ring.

Ideals of a commutative ring (I have only studied the commutative case) are thought of as generalized numbers (in algebraic number theory) and as ring homomorphisms (through the ideal as kernel ...
1
vote
1answer
33 views

Commutative ring is sum of two ideals iff $x \to (x + I, x + J)$ is surjective.

I'm stuck on this exercise and any help would be well appreciated: Let $R$ be a commutative ring with ideals $I,J$. Show that $R=I+J$ if and only if $\phi(x)= (x + I, x + J)$ is surjective from ...
0
votes
3answers
50 views

Showing that $1 + \sqrt{5}$ is irreducible in $\mathbb{Z}[\sqrt{5}]$

Consider the ring $\mathbb{Z}[\sqrt{5}]$. How can we show that the element $1 + \sqrt{5}$ is irreducible in this ring?
1
vote
3answers
61 views

Show that $\mathbb{Z}[\sqrt{5}]$ is NOT a U.F.D.

I'm trying to show that $\mathbb{Z}[\sqrt{5}]$ is not a Unique Factorization Domain. By my understanding, this is the set of all $a + b\sqrt{5}$ such that $a, b \in \mathbb{Z}$. Given this, how can I ...
1
vote
1answer
39 views

relation between units and non zero divisors in a ring

I can prove that in finite commutative ring, non zero divisors are units. My question is if the reverse also true. I mean, units are non zero divisors? And what about the commutative infinite rings?
3
votes
1answer
20 views

$f:R \to D$ a homomorphism of the additive group of rings , $f(aba)=f(a)f(b)f(a) , f(1_R)=1_D$ , then is $f$ a ring homomorphism?

Let $R$ be a ring with multiplicative identity $1$ and $D$ be an integral domain with multiplicative identity ( i.e. $D$ is a commutative unital ring without zero divisors ) , let $f:R \to D$ be a ...
3
votes
2answers
58 views

Can $ℂ$ be viewed as a (nontrivial) field of fractions?

Is there an interesting ring $S ⊂ ℂ$ such that $ℂ = Q(S)$? I’m thinking no, but how can I prove it?
-1
votes
0answers
62 views

Find the number of elements of quotient rings

Let $R$ be the ring obtained by taking the quotient ring of $\mathbb Z_6[x]$ by the principal ideal $(2x + 4)$. Then $R$ has infinitely many elements. we know that $2x + 4 = 0$ $\Rightarrow$ $ x ...
2
votes
0answers
62 views

Localization of euclidean ring is euclidean?

I am trying to prove that a localization of a euclidean ring is euclidean, and the converse statement. I feel the basic definition of the norm is enough but I do not know how. Please note I am very ...
4
votes
2answers
48 views

Question about kernel and homomorphism

I was wondering is there any reason we take the identity e` for the kernel for ring homomorphism to be the additive identity instead of the multiplicative one?
0
votes
1answer
50 views

Nakayama's lemma, second version

Let $R$ be a commutative ring with identity, $J$ an ideal that is contained in every maximal ideal of $R$, and $A$ is finitely generated $R-$ module. If $R/J\otimes _R A=0$, then $A=0$. ...