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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1answer
37 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
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1answer
16 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
28 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
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1answer
35 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 ...
1
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1answer
63 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
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0answers
15 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
6 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 ...
0
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1answer
26 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
23 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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0answers
30 views

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
22 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
33 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
13 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 ...
-2
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0answers
26 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
27 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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0answers
33 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
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0answers
10 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
53 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
41 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
votes
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
votes
2answers
77 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
47 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
69 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
60 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
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4answers
62 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 ...
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0answers
33 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
41 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
23 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 ...
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0answers
38 views

Quotient of Ideals in matrix rings

I'd like to know where could I find some info about the quotient $I:J=\{a\in R\mid aJ\subseteq I\}$ ($R$ a ring) in matrix rings? Or for example, in a matrix ring over $\mathbb{Z}$. I would like to ...
2
votes
1answer
26 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
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1answer
34 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
32 views

Modules over Itself

Let $F$ be a field and let $R$ be the ring of polynomials over $F$ in infinitely many variables $x_1, x_2 , \cdots x_n \cdots$ (Of course, each element of $F$, being a polynomial, will involve only ...
1
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1answer
29 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?
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0answers
51 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
15 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 ...
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2answers
25 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 ...
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2answers
53 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
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0answers
19 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
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3answers
66 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
48 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
56 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 = ...
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0answers
48 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 ...
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0answers
46 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
32 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
32 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
37 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 ...
0
votes
2answers
34 views

Let $I = \{a +\sqrt2b \in \Bbb Z[\sqrt2] : a$ and $b$ are both multiple of $5\}$. [duplicate]

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?
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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 ...
1
vote
1answer
13 views

Commutative rings and ideals, showing a map is well defined

Let $R$ be a commutative ring with an ideal $I$. The additive group $R/I$ is the set of cosets of $I$ with respect to addition in $R$. Let $\cdot : R/I \times R/I \to R/I$ be defined by ...
2
votes
1answer
20 views

Proving that a Left semisimple ring $R$ is both left noetherian and left artinian

Prove that a left semi-simple ring $R$ is both left noetherian and left-artinian. I am following the proof given in pg 27,A first course in non-commutative rings (T.Y.Lam). Its strategy is to show ...