This tag is for questions about rings, which are a type of algebraic structure studied in abstract algebra and algebraic number theory.

learn more… | top users | synonyms (2)

0
votes
0answers
14 views

$\mathbb{Z}[\frac{1-\sqrt{-19}}{2}]$ - principal ring, but not an euclidean ring

I am stucked on the problem. Is there someone who could tell me why $\mathbb{Z}[\frac{1-\sqrt{-19}}{2}]$ is a principal ring, but it is not an euclidean ring?
-1
votes
1answer
27 views

Factoring polynomials in $\Bbb Z_n$

a). Factor $f(x) = x^3 + 4x^2 + 5x + 2$ completely over $\Bbb Z_7$. b). Give two different factorizations of $x^2 + x + 8$ in $\Bbb Z_{10}[x]$. I have found the zeros of both of these but I am ...
0
votes
1answer
47 views

Find all the zeros of $f(x) = x^3 + 3x + 5$ in $\Bbb Z_7$

Find all the zeros of $f(x) = x^3 + 3x + 5$ in $\Bbb Z_7$. I've tried factoring this into multiple forms but I can't seem to find an easy way to find the $x'$s for $x^3 + 3x + 5 = 0$. Any hints or ...
0
votes
1answer
17 views

Two questions regarding polynomial rings.

Give an example of a natural number $n > 1$ and a polynomial $f(x) ∈ \Bbb Z_n[x]$ of degree $> 0$ that is a unit in $\Bbb Z_n[x]$. For this is set $n=2$. So then $f(x) = x \in \Bbb Z_2[x] $. ...
0
votes
2answers
41 views

General questions about Polynomial Rings [on hold]

I'm learning about polynomial rings in my class. My instructor and book are both spectacularly unhelpful and didn't even bother to define most of the terms in my homework. So I have some general ...
1
vote
1answer
16 views

Let R* be the set of units of R and S* be the set of units of S. Prove that f(R*) = S*.

Let R and S be commutative rings with unity $1_R$ and $1_S$ respectively, and let $f: R\to S$ be a ring isomorphism. I am at a loss. Any help is much appreciated.
0
votes
1answer
61 views

Lie rings: reference request

Dear friends: I am looking for a modern reference for Lie rings (In particular, I would like to have nice references for the structure of Lie ideals), let it be lecture notes or a book, in the sense ...
0
votes
2answers
29 views

$f(x) \in \mathbb{R}[x]$. Prove that if $z = a + bi$ is a zero of $f(x)$ then $z = a − bi$ is also a zero of $f(x)$.

Let $f(x) \in \mathbb{R}[x]$. Prove that if $z = a + bi$ is a zero of $f(x)$ then $z = a − bi$ is also a zero of $f(x)$. I'm learning about polynomial rings but my book and my instructor never ...
2
votes
0answers
19 views

$R$ be a commutative ring with unity such that every surjective ring homomorphism $f:R\to R$ is injective , then is $R$ Noetherian?

Let $R$ be a commutative ring with unity such that every surjective ring homomorphism $f:R\to R$ is injective , then is $R$ a Noetherian ring ?
1
vote
1answer
62 views

why $RM \neq 0$

Definition: we said that $M$ is simple if $RM \neq 0$ and $M$ has no proper submodule. i couldnt understand why it must be $RM \neq 0$ could you please explain why ? Thank you for your helping.
21
votes
3answers
902 views

What does the topology on $\operatorname{Spec}(R)$ tells us about $R$?

Let $R$ be a commutative ring with a unit. $\newcommand{\spec}{\operatorname{Spec}}\spec(R)$ denotes the set of all prime ideals in $R$, and it can be topologized using the Zariski topology. Last ...
1
vote
2answers
118 views

Looking for the definition of 'locally finite-dimensional'

Recently, reading the book 'Skew Linear Groups' by M. Shirvani and B. A. F. Wehrfritz, I've encountered the following: ...
1
vote
0answers
12 views

What are the primary submodules of a finitely generated module over a PID ?

How do we calculate i.e. explicitly determine the primary submodules of a finitely generated module over a PID ? And the primary submodules of a finite abelian group ?
8
votes
1answer
74 views

Commutativity of a ring from idempotents.

In a ring $R$ with unity, every element can be written as product of finitely many idempotents. Can one show that the ring is commutative?
0
votes
0answers
31 views

$A,B\in\mathbb Q[x]$ with $A,B$ monic, and $ AB\in\mathbb Z[x]$, prove $A,B\in\mathbb Z[x]$

It is part of cyclotomic polynomials. But I don't know how to deal with it and what to do next. I have prove $n$-th root is related to Euler's totient fuction. But I don't know how to use it. Thank ...
8
votes
3answers
111 views

Any left ideal of $M_n(\mathbb{F})$ is principal

I'm working on the following problem: Let $A$ be the ring of $n \times n$ matrices over a field $\mathbb{F}$. (a) Show that for any subspace $V$ of $\mathbb{F}^n$, the set $I_V$ of matrices ...
1
vote
1answer
45 views

Do we have to show that $f(x)\in R$?

Let $R$ be a commutative ring with unity. I want to show that if $g(x)=c_nx^n+\dots+c_0\in R[x]$ is a zero divisor of $R[x]$ then there exists $d\in R \setminus \{0\}$ such that $dc_n=dc_{n-1}=\dots ...
0
votes
2answers
27 views

$\mathbb F_q[x]/(p(x))$ is a field of order $q^n$.

Let $\mathbb F_q$ be a field of order $q$ and $p(x)$ be an irreducible element in $\mathbb F_q$ of degree $n$. Then prove that $\mathbb F_q[x]/(p(x))$ is a field of order $q^n$. Attempt: As $p(x)$ ...
0
votes
0answers
58 views

Simple algebra that is not a simple ring

Maybe this question is trivial, however I'm not acquainted with non-commutative stuff. Here it's written that a simple algebra may not be a simple ring. The definition of simplicity are the usual ...
-3
votes
1answer
20 views

The mapping defines a unique automorphism [on hold]

Let $R$ be a commutative ring with unity and $a,b\in R$ with $a$ invertible. I want to show that the mapping $x\rightarrow ax+b$ defines a unique automorphism of $R[x]$ that is idempotent in $R$. ...
4
votes
2answers
157 views

Suppose that for every nonzero $R$-ideal $I$ and element $a \in I$ there exists a unique $R$-ideal $J$ such that $IJ=(a)$. Then $R$ is Noetherian.

Suppose that for every nonzero $R$-ideal $I$ and element $a \in I$ there exists a unique $R$-ideal $J$ such that $IJ=(a)$. Then $R$ is Noetherian. I'm having trouble proving this. To note, $R$ is ...
0
votes
2answers
25 views

Why is $\varphi(X_i) = X_i + b_i$ an automorphism of $K[X_1,\dots,X_n]$?

I'm trying to justify to myself the assertion (used here) that given a field $K$ and elements $b_1,\dots,b_n\in K$, the map $\varphi(X_i) = X_i + b_i$ is a $K$-automorphism of $K[X_1,\dots,X_n]$. ...
3
votes
1answer
48 views

Rings in which $ab=0$ implies $axb=0$

I'm sure there must be some standard term for (not necessarily commutative) rings $R$ in which $ab=0$ implies $(\forall x)\, axb=0$ (for example, this is the case if $R$ is commutative or is a ...
1
vote
1answer
49 views

Problems with proof of Krull's height theorem

I want to understand the proof of next Theorem. Let $A$ a Noetherian ring and $\mathfrak a=(a_1,...,a_n)$ a proper ideal of $A$. Let $\mathfrak p\in\mathrm{Spec}(A)$ a minimal ideal over ...
2
votes
1answer
24 views

$R$ be a finite commutative ring such that distinct ideals of $R$ have distinct orders (size) ; is $R$ a PIR?

Let $R$ be a finite commutative ring such that distinct ideals of $R$ have distinct orders (size) ; then is $R$ a Principal ideal ring (PIR) ? What if we moreover assume that distinct subrings of $R$ ...
2
votes
1answer
22 views

Direct Summands of PI Rings as Right Ideals

Is any direct summand of a PI-ring (polynomial identity ring) necessarily idempotent as a right ideal? The answer is yes for a special case of PI-rings, namely any direct summand of a ...
2
votes
1answer
54 views

Is there an adjective for rings whose every non-zero prime ideal is maximal?

(All my rings are commutative and unital.) Question. Is there an adjective for rings whose every non-zero prime ideal is maximal? Remarks: Every PID has this property; more generally, every ...
2
votes
1answer
36 views

Each proper ideal is a product of prime ideals

$R$ is a commutative ring with unity. If $R$ is P.I.D. I want to show that each of its proper ideal is written as a product of prime ideals. $$$$ Since $R$ is a P.I.D. every ideal is a prime ...
2
votes
1answer
31 views

Book recommendation on Primary decomposition of ideals [on hold]

I'm trying to prepare a presentation on "Primary Decomposition of Ideals" which is the title of my project. But I'm new for the subject so I need help on the following points How to outline my ...
1
vote
2answers
32 views

Let $M$ be a non zero maximal ideal in $\mathbb C[x].$ Prove that there exists $a\in \mathbb C$ such that $M=\langle x-a\rangle$.

Let $M$ be a non zero maximal ideal in $\mathbb C[x].$ Prove that there exists $a\in \mathbb C$ such that $M=\langle x-a\rangle$ (ideal generated by $x-a$). Attempt: As $\mathbb C[x]$ is a PID, ...
0
votes
0answers
20 views

If $R$ is a ring and $M$ is a left simple $R$-module, then $R/ann_{R}M$ is a left primitive ring

I'm attempting to prove that if $R$ is a ring and $M$ is a left simple $R$-module, then $R_1=R/ann_{R}M$ is a left primitive ring. I know that this becomes trivial if M is a faithful simple left ...
0
votes
3answers
25 views

Annihilator of rings [duplicate]

If $A$ is an $R$-module, I am having difficulty proving that $A$ is also a well-defined $R/ann(A)$-module with $(r+ann(A))a=ra$.
2
votes
0answers
31 views

What is the intuition behind a Euclidean function?

Many algebra textbooks give the definition of a Euclidean domain as an integral domain $R$ equipped with a Euclidean function/map (let's call it $\nu$). What I don't understand is the significance of ...
-4
votes
1answer
34 views
0
votes
0answers
48 views

Decision between right and left ideals of a ring?

Let us suppose whe find a phenomenon (in nature, social sciences, whatever) for which we believe (or some author has stated) it is possible a formalization or modeling in terms of ideal of a ring. Let ...
0
votes
0answers
9 views

The module of infinite matrices has bases with any length, isn't it? [duplicate]

Let $R$ be a ring. We consider matrices of elements from $R$ with the following properties: The sizes of a matrix is infinite; Any row of a matrix have a finite number of nonzero elements of $R$ ...
1
vote
1answer
52 views

Associativity of the tensor product of bimodules

Let $A_0,\dots,A_n$ be algebras over some fixed commutative ring $k$ (you may assume $k=\mathbb{Z}$ for simplicity). Let $M_i$ be an $(A_{i-1},A_i)$-bimodule for $i=1,\dots,n$. A multilinear map from ...
4
votes
1answer
136 views

$M(A)=\left\{g\in G:g^n=\prod_{i=1}^ma_i^{n_i},a_i\in A,n_i\in\mathbb N\cup\{0\},n\in\mathbb N\right\}?$

Let $G$ be a group and $\emptyset\neq A\subseteq G$. Let $$M(A)=\bigcap_{A\subseteq C}C$$ where $C$ is any subset of $G$ such that $c^k\in C$ for all $c\in C$ and $k\in\mathbb N\cup\{0\}$ and if ...
0
votes
1answer
44 views

Finding the maximal ideals of the quotient of a polynomial ring by an ideal

Let the field $k$ be algebraically closed, let $f(X) \in k[X]$ be a separable polynomial of degree at least $2$, let $$ B = \frac{k[Y,X]}{(Y^2 - f(X))} $$ and write $y,x$ for the images in $B$ of $Y$ ...
17
votes
0answers
147 views
+100

Does every ring of integers sit inside a ring of integers that has a power basis?

Given a finite extension of the rationals, K, we know that $K=\mathbb{Q}[\alpha]$ by the primitive element theorem, so every $x \in K$ has the form $$x = a_0 + a_1 \alpha + ... + a_n \alpha^n$$ with ...
0
votes
0answers
19 views

Ring $R=\{a+b\sqrt{-5} | a,b\epsilon Z $} [duplicate]

Show that in the Ring $R=\{a+b\sqrt{-5} | a,b\epsilon Z $} the element $\alpha$= 3 and $\beta= 1+2\sqrt(-5)$ are relatively prime, but $\alpha \gamma$ and $\beta\gamma$ have no GCD in R, where ...
0
votes
2answers
39 views

Principal Ideal using coordinates?

I thought I understood principal ideals but now im stuck... I want to find the elements of the principal ideal $\langle(1,0)\rangle$ in the ring $\mathbb Z_3\times \mathbb Z_3$ with $+_3$ and $*_3$ in ...
1
vote
1answer
30 views

Finding exact isomorphism between finite fields given as quotient rings [duplicate]

I have two quotient rings over $\Bbb F := GF(3)$: $$\Bbb F[x] / (x^3 -x - 1) \qquad \text{and} \qquad \Bbb F[x] / (x^3 -x + 1) .$$ These things I know: Both quotient rings are irreducible, that means ...
1
vote
0answers
11 views

Can we find ( characterize ) all non-zero commutative Artinian rings $R$ for which $-1 , 1$ are the only units of $R$ ?

Can we find ( characterize ) all non-zero commutative Artinian rings $R$ for which $-1 , 1$ are the only units of $R$ ?
2
votes
3answers
72 views

Showing $\mathbb{Q} \times \mathbb{Q}$ is not a field

I am revising and have come across the question Show that $\mathbb{Q} \times \mathbb{Q}$ with element-wise addition and multiplication is not a field I don't understand how to go about this, do i ...
2
votes
0answers
28 views

Any characterization of rings over which submodules of left free modules are free?

Let $R$ be a ring (with identity, but not necessarily commutative). If given the presumption that for any left free $R$-module $M$, every submodule of $M$ is free, what can we say about $R$? ...
6
votes
3answers
2k views

An ideal that is radical but not prime.

I'm preparing for an exam and, as part of this preparation, I'm looking for an ideal $I$ in an integral domain $R$ that is radical but not prime. Here is an example I'm fooling around with: ...
0
votes
1answer
42 views

describe explicitly all the ideals of $R/(f(x))$

Let $R := \mathbb R[x]$ be the polynomial ring over the real numbers and $f(x) = x^3 - x^2 \in R$. Describe explicitly all the ideals of $R/(f(x))$ where $(f(x))$ is the ideal of $R$ generated by ...
-1
votes
0answers
25 views

Maximal Ideal of ring $C[0,1]$ [duplicate]

Prove that an ideal $M$ of the ring $C[0,1]$ is maximal iff there exists some $a$ in $[0,1]$ such that $M=\{f \in C[0,1]:f(a)=0\}$.
1
vote
2answers
56 views

Proving that $f(x)$ is irreducible in $F[x]$ iff $\phi(f(x))$ is.

This is what I'm proving: Let $F$ be a field. Let $\phi : F[x]\to F[x]$ be an isomorphism such that $\phi(a)=a$ for every $a\in F$. Prove that $f(x)$ is irreducible in $F[x]$ iff $\phi(f(x))$ is. ...