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

$ \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)?
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1answer
19 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.
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2answers
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difference in commutative ring and non-commutative ring

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 ...
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3answers
61 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 ...
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0answers
50 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 $\mathbb{C}[x]$ ...
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1answer
43 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 ...
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2answers
31 views

Integral domain, UFD and PID related problem

(i) Let $R$ be an integral domain that has irreducible elements. Prove that $R[X]$ is not A PID. (ii) Let $R$ be a UFD and $K$ its field of fractions. Let $f \in R[X]$ be a monic polynomial ...
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3answers
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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) = ...
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0answers
66 views

homework: rings, matrices and polynomials

$A,B$ are both $n \times n$ and diagonal matrices. Prove that there is a matrix $X$ which is $n \times n$, and polynomials $p$ and $q$ such that $A= p(X), B= q(X)$ Is this true for ANY 2 matrices (we ...
2
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1answer
45 views

In $\mathbb{Z_2[x]}/(x^2 + x +1)$, what is $\overline{x} \cdot \overline{x}$?

In $\mathbb{Z_2[x]}/(x^2 + x +1)$, what is $\overline{x} \cdot \overline{x}$? $\overline{x} \cdot \overline{x} = \overline{x^2}$ If I divide $\overline{x^2}$ by $(x^2 + x +1)$ I get an answer of $1$ ...
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23 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 ...
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1answer
308 views

Example of a ring satisfying this variant definition of “symmetric” on nilpotent elements

I want an example to show that if $a,b$ are nilpotent elements of a ring $R$ with 1 and if $c$ is any element of $R$, then $abc=0\Rightarrow acb=0$ but $cab=0$ does not imply $acb=0$. This is unlike ...
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2answers
55 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 ...
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2answers
37 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 ...
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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 = ...
3
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2answers
43 views

Proving equivalent versions of faithfully flatness.

I was reading a proof of the the following theorem from Matsumura (p.47) There was something confusing about $(3) \implies (2)$ and $(2) \implies (1)$. Question 1 Here, it says $M \not= ...
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2answers
64 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 ...
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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?
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1answer
26 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 ...
1
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1answer
38 views

If A , B are finitely generated R-algebras then $A\otimes_RB$ is a finitely generated $R$-algebra.

$A$, $B$ are finitely generated $R$-algebras. $R$ is a commutative ring with $1$. Then how can I show that $A\otimes_RB$ is finitely generated $R$-algebra? What I have tried: First I have to show ...
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1answer
35 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$, $\mathbb{Z}[i\sqrt{5}]$, $P[x]$ ...
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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 ...
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2answers
166 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 ...
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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 ...
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1answer
55 views

Proving $M$ is maximal if the quotient ring $R/M$ is a field.

Let $R$ be a ring with unit element and ideal, $M$, such that $R/M$ is a field. Prove $M$ is maximal ideal. I know that because $R/M$ is a field, its only ideals are $(0)$ and itself. Also, I ...
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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) $0_s \in f(I)$ ...
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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 ...
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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 ...
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2answers
32 views

Clarification on quadratic ring notation

My Abstract Algebra text is using the notation $\mathbb{Z}[1 + \sqrt{-5}]$ and calling it a "quadratic integer ring." Just to clarify, $\mathbb{Z}[1 + \sqrt{-5}]$ is simply the set $$ \left\{ a + b(1 ...
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2answers
50 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 = ...
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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 ...
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2answers
49 views

Proving Fermat's Little Theorem in general and use that to prove Euler's Generalization of Fermat's Little Theorem

Can anyone help me with this? I know there are many different ways to do this and threads explaining this question. However I can't seem to find one that uses only group/ring theory. I haven't ...
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4answers
81 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 ...
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2answers
128 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 ...
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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 ...
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1answer
26 views

Maximal and prime ideals of $2 \mathbb Z$

What are the all maximal ideals of $2 \mathbb Z$ ? what are the all prime ideals of $2 \mathbb Z$ ? We know that if $R$ is a commutative ring with multiplicative identity $1$ and $M$ is a maximal ...
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1answer
47 views

Does there exist a non-commutative ring of order $210$? [closed]

Does there exist a non-commutative ring of order $210$? One can construct rings of this order by taking products of $\mathbb Z/n\mathbb Z$, but those are commutative. Historical aside (not ...
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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 ...
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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 ...
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1answer
44 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 = ...
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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 ...
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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')\\ ...
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1answer
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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 ...
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2answers
481 views

Does localization preserve reducedness?

Is the localization of a reduced ring (no nilpotents) still reduced?
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3answers
57 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 ...
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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?
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1answer
83 views

Prove that $S$ is an integral domain and $T$ is not an integral domain.

Let $R = \mathbb{C}[x,y]$ $R^i \subset R$ be the abelian subgroup of $R$ generated by elements of $\mathbb{C}$ times monomials of degree at least $i$ $I = (x^3+x^2-y^2)$ $S = R/I$ $S^i$ be the group ...
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1answer
54 views

Quotient of polynomial ring in two variables by $(1-xy)$ is a PID

With $K$ a field and $K[x,y]$ the polynomial ring over it in two variables, the quotient ring of it over the ideal generated by $1-xy$ is a PID. I've tried using Noetherianess but haven't gotten ...
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1answer
38 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?
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1answer
65 views

Questions about the ring $R=(\Bbb Z/6\Bbb Z)[X]/(2X+4)$ [closed]

Let $R=(\Bbb Z/6\Bbb Z)[X]/(2X+4)$. Then A. $R$ has infinitely many elements? B. $R$ is a field? C. $5$ is a unit in $R$? D. $4$ is a unit in $R$? Can any one help me ..thanks for your time yes ...