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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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
32 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 = ...
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2answers
33 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
32 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
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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) $0_s \in f(I)$ ...
0
votes
1answer
62 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
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1answer
29 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
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 ...
0
votes
1answer
24 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
29 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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2answers
49 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
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3answers
60 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
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4answers
79 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
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1answer
36 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
122 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
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2answers
164 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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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 ...
1
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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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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
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2answers
41 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
43 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')\\ ...
2
votes
2answers
70 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
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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?
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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 ...
1
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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?
3
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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?
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0answers
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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
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0answers
61 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
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2answers
46 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?
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1answer
49 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 $A=0$. ...
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0answers
26 views

Noetherian local ring with maximal ideal $M$

Let $R$ be a Noetherian local ring with maximal ideal $M$. If the ideal $M/M^2$ in $R/M^2$ is generated by $\{ a_1+M^2, \dots, a_n +M^2\}$, then the ideal $M$ is generated in $R$ by $\{ a_1, \dots , ...
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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
45 views

Confused on notions of maximal ideal and some notation

I'm just getting started learning ring theory and am currently learning about ideals. By book (Dummit & Foote) says the following: For example, in the ring $R = \mathbb{Z}[x]$ the elements $2$ ...
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1answer
20 views

Isomorphism between modules over a semisimple ring

If $P$ is a module over the semisimple ring $R/J$, where $R$ is a semilocal ring having $1$, and $J$ is its Jacobson radical, does any isomorphism $P⊕...⊕P≅P'⊕...⊕P'$ with the same (finite) number of ...
0
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0answers
14 views

From progenerators to progenerators

I know that if $R$ is a ring (with identity) and $P$ is a progenarator right $R$-module (a f.g. projective generator) then $P/PJ$ is clearly f.g. when $J$ is the Jacobson radical of $R$; but, how ...
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1answer
90 views

Rings where every subgroup of the additive group is an ideal?

The rings $\mathbb Z$ and $\mathbb Z/n\mathbb Z$ have the property that every subgroup of the additive group is also an ideal (i.e., every subgroup absorbs multiplication by all ring elements). This ...
1
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1answer
27 views

one to one correspondence of Ideals in a ring and its localization

Let $A$ be a commutative ring, and $S$ a mutiplicatively closed subset. In my text book, it is stated that: there is one to one correspondence of prime ideals in ring $A$ (not meeting $S$) and ...
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0answers
30 views

A question about a system of congruent equations? Is there a unified proof by using ring theory?

In his book ``Topics in number theory, Volumes I and II''. William J. Leveque proved the following theorem(see page 34) Theorem A necessary and sufficient condition that the system of congruences ...
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2answers
71 views

Ring whose all ideals are double-sided is commutative?

I was thinking about the following problem: Suppose R is a ring s.t. every left ideal is also right. Is R commutative? This actually continues the easier question: Suppose G is a group whose ...
0
votes
1answer
40 views

Intersection of distinct maximal ideals in a commutative ring with identity.

If $R$ is a commutative ring with identity and $M_1, \dots, M_r$ are distinct maximal ideals in $R$, then show that $M_1\cap M_2 \cap \cdots \cap M_r = M_1M_2\cdots M_r$. Is this true if "maximal" is ...
1
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0answers
20 views

$a=3X^2+X+2 \in \mathbb{Z}_7[X]$. Compute the inverse of $[a]$ in $\mathbb{Z}_7[X]/(X^3+4)$

$a=3X^2+X+2 \in \mathbb{Z}_7[X]$. Compute the inverse of $[a]$ in $\mathbb{Z}_7[X]/p(x)$ where $p(x)=X^3+4$ I know that we want some $[b]\in\mathbb{Z}_7[X]/p(x)$ such that $[a][b]=[1]$. I used ...
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1answer
55 views

If a=b, how does a-b=0?

Let R be a ring and let $a,b \in R$. Suppose $a=b$. How do you know that $a +- b=0$? In other words, what axiom or step lets you say that $a +-b = b + -b$? (That you can add the same thing to both ...
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0answers
41 views

Two properties related to semisimple rings

Let $R$ be a semisimple ring Show the following (i) If $xy=1 \in R$, then $yx=1$. (ii) If $x \in R$ is such that $xR$ is a left ideal of $R$, then $xR=Rx$. I am pretty lost with the two items. I ...
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1answer
41 views

Finite Extension of Integral Domains.

Let $D\subset E$ (integral domains), with fraction fields $k\subset K $. Suppose that $E$ is integral over $D$, and $E$ is $D$-module finitely generated. My question is: $[K:k]$ is finite? Thank ...
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2answers
361 views

Is there an easy way to remember the ring axioms?

I'm working on a problem that asks me to prove a set is a ring. I have to look up the axioms to prove it... I was curious, is there an easy way to remember the ring axioms, so that you can avoid ...
2
votes
1answer
24 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 ...
0
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1answer
15 views

$m,n>1$ are relatively prime integers , then are there at-least four idempotent (w.r.t. multiplication) elements in $\mathbb Z_{mn}$ ?

If $m,n>1$ are integers such that $g.c.d.(m,n)=1$ then is it true that there are at-least four elements in $\mathbb Z_{mn}$ such that $x^2=x$ ( i.e. idempotent ) ?