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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Show that if $R$ and $S$ are ideal of a ring $A$ then the product $R\cdot S$ is a ideal of $A$. [duplicate]

How to prove that if $R$ and $S$ are ideal of a ring $A$ then the product $R\cdot S$ is a ideal. I can't show only that if $x, y\in R\cdot S$ then $x-y\in R\cdot S$. The other axioms of ideal I ...
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3answers
27 views

Example of ideal generated by two elements

I have an easy example on my notes that I don't understand. My teacher said that in $\mathbb{Z}$, $(2,3)=2\mathbb{Z}+3\mathbb{Z}$ is a principal ideal, because $2\mathbb{Z}+3\mathbb{Z}=\mathbb{Z}$. ...
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1answer
17 views

Is group of units of a polynomial ring only constant polynomial which is involved in R

Let R be a integral domain(or maybe field) edit : Let R be a field. The group of units of R[x] is $$ a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \cdots+ a_2 x^2 + a_1 x + a_0 $$(or infinity) such ...
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0answers
23 views

Commutative ring of prime power order

Suppose $R$ is a non trivial commutative ring with identity of prime power order. What can we say about the structure of $R$? If $R$ is of prime order, then $R$ is a field?
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14 views

$x^{mn} -a$ is irreducible in F[x] iff $x^m -a$ and $x^n -a$ are irreducible.

Let F be any field, a is in F and (m,n)=1. Show that $x^{mn}-a$ is irreducible in F[x] iff $x^m -a$ and $x^n -a$ are irreducible in F[x]?
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0answers
29 views

if the sum of two units is a unit, then there is an unique maximal ideal

Let $R$ be a ring with identity element. I have to proof that if the sum of two units of $R$ is a unit, then $R$ has an unique maximal ideal. But i don't see a connection. If someone could give me a ...
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23 views

For a field $K$ is $K\subset{K[X_{1},…X_{n}]}$

Let $K$ be any field and $K[X_1,...X_n]$ the ring of polynomials in $X_1,...X_n$ with coefficients in $K$. I am wondering if $K$ is a subset of $K[X_1,...X_n]$. I believe $K\subset{K[X_1]}$ since ...
2
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1answer
14 views

Prove that $R$ has ACCP if and only if every non-empty collection of principal ideals of $R$ has a maximal element.

Let $R$ be a commutative ring. Prove that $R$ has ACCP if and only if every non-empty collection of principal ideals of $R$ has a maximal element. Prove further that if $R$ is an integral domain and ...
1
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1answer
17 views

The norm of Gaussian integers and the irreducible element $ 1 + i $.

Note: Let $ \text{N}(a + bi) \stackrel{\text{df}}{=} a^{2} + b^{2} $. Observe that $ \text{N}(1 + i) = 2 $. Is it always true that if $ 1 + i $ divides a Gaussian integer, then the norm of $ 1 + i $ ...
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1answer
25 views

Quotient of the ring of integers of a quadratic field by the ideal generated by a split integer prime.

I am wondering about primes $p$ in $\mathbb Z$ that are split in $\mathcal O_{K}$, $K=\mathbb Q(\sqrt d)$. Let $\omega=\sqrt d$ if $d \equiv 2,3 \mod 4$ and $\omega=\frac{1+\sqrt d}{2}$ if $d \equiv 1 ...
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2answers
69 views

What are all the integral domains that are not division rings?

A commutative division ring is an integral domain. But what are all the integral domains that are not division rings? The examples I currently know are the following: $\mathbb{Z}$, $\mathbb{Z}[i]$, ...
1
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1answer
19 views

How to prove subfield generated $K(u_1,u_2,..u_{n-1},u_n)=K(u_1,u_2,..u_{n-1})(u_n) $

This is problem in Hungerford chapter 5: Fields and Galois Theory. Prove $K(u_1,u_2,..u_{n-1},u_n)=K(u_1,u_2,..u_{n-1})(u_n)$ and $K[u_1,u_2,..u_{n-1},u_n]=K[u_1,u_2,..u_{n-1}][u_n] $ My ...
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1answer
22 views

Show that $m = \pm 2$ or $m = \pm 3.$

Let $$R = \left\{\frac{a + b\sqrt{-19}}{2}:a,b \in \mathbb{Z}, a \equiv b \mod 2 \right\} = \mathbb{Z} \left[\dfrac{1+\sqrt{-19}}{2} \right] = \mathbb{Z}[\alpha].$$ and define $d:R \setminus \{0\} ...
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2answers
22 views

Let $q$ be a prime congruent to 3 mod 4, prove the quotient ring $\mathbb{Z}[i]/(q)$ is a field with $q^2$ elements

Let $q$ be a prime congruent to 3 mod 4, prove the quotient ring $\mathbb{Z}[i]/(q)$ is a field with $q^2$ elements The field portion I understand. $\mathbb{Z}[i]$ is a PID and because $q$ is ...
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0answers
37 views

Let $\mathbb{Z}[i]$ denote the Gaussian integers. The set of units of $\mathbb{Z}[i]$ is $\{\pm 1, \pm i\}.$

Let $\mathbb{Z}[i]$ denote the Gaussian integers. The set of units of $\mathbb{Z}[i]$ is $\{\pm 1, \pm i\}.$ A proof from: https://proofwiki.org/wiki/Units_of_Gaussian_Integers (Proof 2) Proof. ...
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1answer
35 views

Prove that if $z$ is good then so is $z + r$ for every $r \in R$.

Let $$R = \left\{\frac{a + b\sqrt{-19}}{2}:a,b \in \mathbb{Z}, a \equiv b \mod 2 \right\} = \mathbb{Z} \left[\dfrac{1+\sqrt{-19}}{2} \right] = \mathbb{Z}[\alpha].$$ Note that $R$ is an integral ...
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1answer
34 views

Show that if $r$ is nilpotent in a ring with identity, then $1-r$ is a unit in $R$ [duplicate]

Let $R$ be a ring. An element $r \in R$ is called nilpotent if $r^n=0$ for some integer $n \ge 1$. Show that if $r$ is nilpotent in a ring with identity, then $1-r$ is a unit in $R$. Proof. Recall ...
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2answers
36 views

What is $Q(x)$?

I do not really understand what $\mathbb{Q}(\pi)$ is here: Ofcourse we see that $\mathbb{Q}(\pi)$ is a field. But I have to "guesses" of what they mean, is one of them correct? 1. ...
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29 views

Let $R$ be a ring with identity. Prove that if $1-ab$ is invertible for some $a,b \in R$, then $1-ba$ is also invertible. [duplicate]

Let $R$ be a ring with identity. Prove that if $1-ab$ is invertible for some $a,b \in R$, then $1-ba$ is also invertible. Ok, si if $R$ is a ring with unity, then we have $R$ with $1 \ne 0$ We have ...
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0answers
12 views

Direct product of Rings isomorphism

I was reading chapter five in my Abstrat algebra book about finite Abelian groups. In Proposition 6 part (1). It states that $Z_{m} \times Z_{n} \cong Z_{mn}$ if and only if $gcd(m,n)=1$. This ...
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1answer
36 views

Is $f(x)$ reducible if $f(a)=0$

I am confused about this seemingly trivial question: If $f(a) = 0$ for some $a\in D$, then when is $f(x)$ reducible in $D[x]$? ($D$ is an integral domain). My answer: Always. Let $f(a)=0$. ...
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0answers
38 views

Find elements in the center of $n × n$ matrix ring $M_n (R) $ for any $n ≥ 2$. [duplicate]

Let $R$ be a ring. The center of $R$ is the set $C(R) = \{c ∈ R : cr = rc, ∀r ∈ R\}$. Determine elements in the center of the $n × n$ matrix ring $M_n (R) $ for any $n ≥ 2$. So, we have that ...
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2answers
44 views

Uniqueness of prime ideals of $\mathbb F_p[x]/(x^2)$

What are the prime ideals of $\mathbb F_p[x]/(x^2)$? I have been told that the only one is $(x)$, but I would like a proof of this. I want to say that a prime ideal of $\mathbb F_p[x]/(x^2)$ ...
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Semiprime group rings

$R$ is commutative semiprime ring, $(R,+)$ abelian group without torsion. Then $RG$ is semiprime. Proof by contradiction. If $x$ is nilpotent element in $RG$, then $x=r_1g_1+r_2g_2+...r_ng_n$. ...
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1answer
24 views

Factoring the Ring of Integers into Ideals

Let $K$ be a number field. Let $\frak p$ be a prime ideal in $\mathcal O_K$. Let $u\in \mathcal O_K$ and $m\in \mathbb N$. I've been told that $|u|_{\frak p} = |m|_{\frak p} = 1$ where $|\cdot|_{\frak ...
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1answer
95 views
+500

Fermat's last theorem and $\mathbb{Z}[\xi]$

I heard that one can prove special cases of FLT by using unique factorization in $\mathbb{Z}[\xi]$ (whenever this is possible), where $\xi$ is a primitive $n$-th root of unity. How can one do this in ...
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1answer
20 views

Showing $\hat{A} \otimes_{A} M \cong \hat{M}$ when $M$ is a finitely generated free $A$-module.

I had a reading question on Proposition 10.13 from Atiyah-MacDonald. The proposition is the following PROPOSITION. For any ring $A$, if $M$ is finitely-generated, $\hat{A} \otimes_{A} M \rightarrow ...
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1answer
33 views

Set of units in ring a group?

I am supposed to prove that given a commutative ring $R$, the set of units $R^{\times}$ is a group. I checked the axioms of a group and it all came down to noting that if $a,b\in R^{\times}$, then ...
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1answer
65 views

Nilpotent elements in group algebra

Suppose $FG$ -- is group algebra and $F$ is field with characteristic $p>0$. $G$ - is finite $p$-group. Thus, it's clear that $(e-g)$ is nilpotent. But how to show that $(e-g)g_1$ is nilpotent for ...
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0answers
91 views

How to prove that an ideal can not be generated by 2 elements

In Kunz's "Introduction to commutative algebra and algebraic geometry", page 137-139, particular monomial affine curves are described. Here is the link. In case the curve is not an ideal ...
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1answer
25 views

Example of a domain where all irreducibles are primes and that is not a GCD domain

One has the following relations for a domain $R$: $R$ GCD domain $\Rightarrow$ All irreducible elements are prime $R$ PID $\Rightarrow$ $(R$ GCD domain $\land$ $R$ statisfies ACCP$)$ $R$ UFD ...
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0answers
9 views

$U_1(\mathbb{Z}G)$ is a finitely generated FC-group.

If each member in support of an element in $\mathbb{Z}G$ is centralized by a subgroup of finite index in $G$, then why does it imply that $U_1(\mathbb{Z}G)$ is a finitely generated FC-group., where ...
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1answer
32 views

how to show that an ideal is convex [on hold]

I need to show that the ideal $J=(i)$ in $C(\mathbb R)$ where $i$ is the identity function, $C(\mathbb R)$ is the ring of all continuous functions on the real numbers, is a convex ideal.
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1answer
29 views

What are some examples of principal, proper ideals that have height at least $2$?

Krull's principal ideal theorem states that in a Noetherian ring $R$, any principal proper ideal $I$ has height at most $1$. Presumably the Noetherian hypothesis is required, so what are some ...
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If $A$ is a submodule of $B$ and $B$ is a submodule of $C$, is $A$ a submodule of $C$?

Let $C$ be a commutative ring (with 1, if this matters). If $A$ is a submodule of $B$ and $B$ is a submodule of $C$, is $A$ a submodule of $C$? I can't really prove that it is true because it is ...
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0answers
24 views

Integral domain and ideal of ring [on hold]

Let $R$ be an integral domain and $I$ and $J$ be two ideals of $R$ such that $IJ$=$I \cap J$. Show that $R$ is a field.
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1answer
39 views

Problems with understanding the proof of noetherian ring

If $M$ is an $R$-module, the the following are equivalent: 1. M is finitely generated 2. M satisfies the ascending chain condition 3. Every non-empty set of submodules of M contains at least one ...
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0answers
15 views

Does $U=U_1(\mathbb{Z}G)$ normalize $G$?

Let $G$ is an arbitrary group and and $U=U_1(\mathbb{Z}G)$ is the set of normalized units of $ZG$ i.e. $U_1(\mathbb{Z}G)$ here means units of $\mathbb{Z}G$ with augmentation $1$, i.e. set of all ...
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0answers
27 views

a question about abstract algebra,prove that the ring is commutative. [duplicate]

(1)A ring R is a booleean ring if for every $a\in R$,$a^2=a$. Show that every Boolean ring is a commutative ring. (2)Let R be a ring,where $a^3=a$ for all $a\in R$.Prove that R must be a commutative ...
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0answers
31 views

Let $R$ be an Euclidean ring. Prove that $\operatorname{lcm}(a,b) = \dfrac{ab}{\gcd(a,b)}$ for $a,b \in R.$

Let $R$ be an Euclidean ring. Prove that $\operatorname{lcm}(a,b) = \dfrac{ab}{\gcd(a,b)}$ for $a,b \in R.$ Does this proof work? Any criticism appreciated: We rewrite $\operatorname{lcm}(a,b) = ...
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40 views

The difference between the ring version and module version of Chinese Remainder Thereom.

Chinese Remainder Theorem for Commutative Rings If $R$ is a commutative ring with $1$ and $I, J$ are ideals of $R$ that are pairwise coprime or comaximal (meaning $I + J = R$), then $IJ = I \cap J$, ...
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0answers
11 views

Criterions for $U_1(\mathbb{Z}G)=G$ i.e. units to be trivial in $\mathbb{Z}G$

Notation- $U_1(\mathbb{Z}G)$ here means units of $\mathbb{Z}G$ with augmentation $1$, i.e. $u=\sum_{g\in G} a(g)g \in U(\mathbb{Z}G)$ such that $\sum_{g \in G} a(g)=1$ 1) I have done theorem by ...
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1answer
56 views

Is the zero ideal $\{0_{M_2(\mathbb{R})}\}$ maximal in $M_{2}(\mathbb{R})$?

Consider the ring $M_2(\mathbb{R})$ of all $2 × 2$ matrices over $\mathbb{R}$. Is the zero ideal $\{0_{M_2(\mathbb{R})}\}$ maximal in $M_{2}(\mathbb{R})$? We know, in the ring $\mathbb{Z}$, ...
2
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2answers
47 views

R is commutative ring with identity & define $\circ$ on $R$ by for any $a,b \in R$ $a \circ b=a+b-ab$ Prove the following

Let R be a commutative ring with identity. Define a new operation $\circ$ on $R$ by for any $a,b \in R$ $$a \circ b=a+b-ab$$ a) Prove that $\circ$ is associative b) Prove that R is a field iff the ...
0
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1answer
16 views

Show that $\bar{a}_{n}(\bar{x})^n+···+\bar{a}_{1}\bar{x}+\bar{a}_{0}=0_{F[x]/I}$

Let $F$ be a field, $f(x)$ be an irreducible polynomial in $F[x]$ and $I =(f(x))$. Let $f(x)= a_nx^n+···+a_1x+a_0, a_i \in F$ for $i=0,...,n$. And, $\bar{x} = x + I ∈ F[x]/I$ and $\bar{a_i} = a_i + I ...
6
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1answer
84 views

Let R be a ring. Prove that if $x^2=x$ for each $x \in R$, then R is a commutative ring. [duplicate]

Let R be a ring. Prove that if $x^2=x$ for each $x \in R$, then R is a commutative ring. Ok, so I'm just looking for some confirmation that I'm doing this correctly. If we suppose $x,y \in R$ Let's ...
2
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1answer
48 views

Algebraic and transcendental functions over $\mathbb Z_n$ — is this a known result?

I have proved a result that seems (to me) interesting, and I am wondering whether it is a known result, and if not whether it seems interesting to others. The result is as follows: Let $R=\mathbb Z ...
5
votes
1answer
74 views

If $R/I \times R/J$ is isomorphic to $R/(I\cap J)$ as $ R $-modules, then $I + J = R$. [duplicate]

If $R$ is a commutative ring with identity and $I$ and $J$ are ideals of $R$ such that $R/I \times R/J$ is isomorphic to $R/(I\cap J)$ as $R$-modules, then $I + J = R$. I know this is the ...
4
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3answers
61 views

Example of a ring where all but two of its elements are units

One way of viewing a field is just as a ring where all but one of its elements (namely $0$) is a unit. I'm looking for rings (commutative with a 1) where all but two of its elements are units. I found ...
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4answers
66 views

Characteristic of a Finite Integral Domain

I am a little confused as how to approach this problem. The title of this problem is the title of the section which it comes from. However, there is no information that the given integral domain is ...