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
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0answers
33 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 ...
0
votes
0answers
14 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 ...
6
votes
1answer
39 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 $f(a)=0$. ...
0
votes
0answers
45 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 ...
5
votes
2answers
64 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 $\mathbb F_p[x]/(x^2)$ ...
1
vote
0answers
51 views

Group Ring over commutative ring

$R$ is commutative semiprime ring, $(R,+)$ abelian group without torsion. Then $RG$ is semiprime. Proof by contradiction. If $x$ is nilpotent element in $RG$, then $x=r_1g_1+r_2g_2+...r_ng_n$. ...
1
vote
1answer
28 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 ...
9
votes
2answers
302 views

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 ...
0
votes
1answer
23 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 ...
3
votes
1answer
42 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 ...
0
votes
1answer
71 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 ...
2
votes
0answers
98 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 ...
2
votes
1answer
31 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 ...
0
votes
1answer
41 views

how to show that an ideal is convex [closed]

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.
0
votes
0answers
59 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 ...
1
vote
1answer
43 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 ...
1
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0answers
17 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 ...
0
votes
0answers
32 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) = ...
7
votes
0answers
48 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 $IJ = I \cap J$, ...
0
votes
1answer
21 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 ...
2
votes
1answer
60 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
votes
2answers
50 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 ...
1
vote
1answer
22 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
votes
1answer
90 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
votes
1answer
52 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
83 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
votes
3answers
70 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 ...
0
votes
4answers
76 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 ...
2
votes
0answers
48 views

A quick way to determine number of ring homomorphism

Find the number of ring homomorphism from $\mathbb Z_2\times \mathbb Z_2$ to $\mathbb Z_4$. I know that checking idempotent elements the number of ring homomorphisms is $1$. Again the number of ring ...
3
votes
2answers
43 views

If $x = a +bi$ is not a unit of $\mathbb{Z}[i],$ prove that $a^2+b^2>1.$

A problem from my algebra text: If $x = a +bi$ is not a unit of $\mathbb{Z}[i],$ prove that $a^2+b^2>1.$ I think it's false since $x = 0 + 0i = 0 \in \mathbb{Z}[i]$ is not a unit, but $0 + 0 ...
0
votes
1answer
25 views

A doubt in a lemma on integral group rings.

In a paper by Farkas, I was doing this lemma, where I had this doubt (red underlined) in the proof of the lemma. Can anybody explain me how does it follow $\alpha$ is centralized by $H$. It should ...
2
votes
1answer
27 views

How to classify the rings of fractions of a principal ideal domain?

Let $A$ be a principal ideal domain and let $K$ be its field of fractions. I proved a) Every ring $B$ such that $A \subset B \subset K$ is a ring of fractions of $A$, and c) Show that any ring of ...
1
vote
1answer
36 views

Prove that $\Bbb{Z}[i]/I$ is finite where I is an ideal of $\Bbb{Z}[i]$

Show that for any nontrivial ideal $I$ of $\Bbb{Z}[i]$, $\Bbb{Z}[i]/I$ is finite. $\Bbb{Z}[i]$ is a PID, so $I=\langle{a+ib\rangle}$. Now $\Bbb{Z}[i]/I$ has elements of the form ...
0
votes
1answer
36 views

Modules over Itself

Let $F$ be a field and let $R$ be the ring of polynomials over $F$ in infinitely many variables $x_1, x_2, \dots, x_n, \dots$. (Of course, each element of $R$, being a polynomial, will involve only ...
1
vote
1answer
30 views

When is $\mathbb{Z}[\sqrt{d}]$ an Euclidean domain?

Where $d \in \mathbb{Z}$ is not a perfect square. This problem appeared in our exam and now I'm asking how was I supposed to answer?
0
votes
0answers
58 views

Prove $\mathbb{Z} [\sqrt{2}]$ is a Euclidean ring.

The definition of Euclidean ring: An integral domain R is called Euclidean ring if $\exists \delta$ : $R${$0$} -> $\mathbb{N} \cup{0}$ satisfying: (1) $\delta (a) \leqslant \delta (ab)$ if a, b $\in ...
0
votes
1answer
22 views

$A$ prime in $S$ implies that $\phi^{-1}(A)$ prime in $R$ ; $A$ maximal in $S$ implies that $\phi^{-1}(A)$ maximal in $R$

Suppose $R,S$ are commutative rings with unities. Let $\phi$ be a ring homomorphism mapping $R\to S$ and let $A\subset S$ be an ideal. How can I start the proofs for: Showing that $A$ prime in ...
1
vote
2answers
29 views

If $R$ is a ring with unit element $1$ and $\varphi$ is a homomorphism of $R$ onto $R'$, prove that $\varphi(1)$ is the unit element of $R'$.

Herstein 3.4.20: If $R$ is a ring with unit element $1$ and $\varphi$ is a homomorphism of $R$ onto $R'$, prove that $\varphi(1)$ is the unit element of $R'$. I don't understand why $\varphi$ needs ...
0
votes
2answers
54 views

Show that $Im(\phi) = \mathbb{Z}[i]$

Let $\phi: \mathbb{Z}[x]\to \mathbb{C}$ and $\phi(f(x)) = f(i), \forall f(x) \in \mathbb{Z}[x].$ Show that $Im(\phi) = \mathbb{Z}[i]$ My attempt: I am not sure if it's correct: First, we need to ...
0
votes
0answers
21 views

Jacobson radical of polynomial ring [duplicate]

Let $R$ be a ring, i want to show that: if R has not nil-ideals than $J(R[x]) = \left\{ \emptyset \right\}, \text{where $J$ Jacobson radical, $R[x]$ - polynomial ring over $R$}$
3
votes
3answers
70 views

Is $ \langle x,5 \rangle $ a maximal ideal of $ \mathbb{Z}[x] $?

Here, $ \langle x,5 \rangle $ is the ideal generated by $ x $ and $ 5 $ in $ \mathbb{Z}[x] $, which is the polynomial ring over $ \mathbb{Z} $. How should I approach this question?
3
votes
1answer
94 views

Let $a$, $b$, and $c$ be elements of a commutative ring, and suppose $a$ is a unit. Prove that $b$ divides $c$ if and only if $ab$ divides $c$.

Let $a$, $b$, and $c$ be elements of a commutative ring $R$, and suppose $a$ is a unit. Prove that $b$ divides $c$ if and only if $ab$ divides $c$. Okay so here is the proof I came up with. Please be ...
0
votes
2answers
63 views

Let $R$ be a finite ring with unity. Prove that $x$ is a LZD $\iff$ x is a RZD

Let $R$ be a finite ring with unity. Let $x \in R$. Prove that $x$ is a Left Zero Divisor $\iff$ x is a Right Zero Divisor. My attempt Suppose $x$ is a LZD. Then, $\exists y \in R$ such that $xy = ...
1
vote
0answers
49 views

A ring with a left cancellable element and a right identity always has an identity.

Let $R$ be a ring with $a, e \in R$ such that $a$ is not a left zero-divisor and $be=b, \forall b \in R$. Prove that $R$ has an identity. My attempt Let, $aeb = ab \Rightarrow aeb - ab = 0 ...
1
vote
0answers
49 views

Is the ring of germs of $C^\infty$ functions at $0$ Noetherian?

I'm considering the property of the ring $R:=C^\infty(\mathbb R)/I$, where $I$ is the ideal of all smooth functions that vanish at a neighborhood of $0$. I find that $R$ is a local ring of which the ...
2
votes
1answer
39 views

To prove that $I = \{\,(n,m) \in \Bbb Z \times \Bbb Z : n,m $ are even $\}$ is not a maximal ideal of $ \Bbb Z \times \Bbb Z $.

To prove that $I = \{\,(n,m) \in \Bbb Z \times \Bbb Z : n,m $ are even $\}$ is not a maximal ideal of $ \Bbb Z \times \Bbb Z $. Thus there exists an ideal $J$ of $ \Bbb Z \times \Bbb Z $ such that $I ...
2
votes
1answer
36 views

Ring Homomorphism from $\mathbb{Z}_m$ to $\mathbb{Z}_n$

Suppose $R$ is a ring homomorphism from $\Bbb{Z}_m$ to $\Bbb{Z}_n$ , prove that if $R(1) = a$ then $(a^2)=a$. Also show, its converse is not true. The first part goes like this : $R(1) = a , ...
2
votes
1answer
43 views

Proving every element of $\mathbb{F}[x]/(p(x))$ can be expressed uniquely

Proving every element of $\mathbb{F}[x]/(p(x))$ can be expressed uniquely in the form $a(x) + (p(x))$ where $\text{deg}(a) < \text{deg}(p)$ this is a homework problem and I'm stuck, here is my ...
-1
votes
2answers
83 views

Let $I = \{a +\sqrt2b \in \Bbb Z[\sqrt2] : a$ and $b$ are both multiple of $5\}$. Show that $I$ is a maximal ideal. [closed]

Let $I = \{a +\sqrt2b \in \Bbb Z[\sqrt2] : a$ and $b$ are both multiple of $5\}$. Show that $I$ is a ideal. (I have done it) But how to show that it is maximal?
1
vote
3answers
33 views

Elements of $\mathbb{Z}/(n)$

Let $(n) = \{ \lambda n | \lambda \in \mathbb{Z} \}$. In my book it has shown that every element in $\mathbb{Z}/(n)$ can be expressed uniquely in the form $r + (n)$ where $0 \leq r \leq n-1$ now I ...