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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2
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1answer
53 views

Prove that $p(x)=(x-a)q(x)$ for some $q(x)\in R[x]$ of degree= (degree of $p)-1$.

Let $R$ be a finite commutative ring with identity. Let $a\in R$ is a root of $p(x)\in R[x]$ . Prove that $p(x)=(x-a)q(x)$ for some $q(x)\in R[x]$ of degree= (degree of $p)-1$. If $R$ were a ...
0
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1answer
34 views

A probable equality between two ideals

Let $I$ and $J$ be ideals of a ring $R$. I want to know whether the ideal $(I+J)^2$ equals $I^2+IJ+JI+J^2$. By taking elements and using the definition of product of two ideals $I$ and $J$ as the set ...
1
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1answer
41 views

In what sense are complete local rings finitely generated modules?

In the first paragraph of section 18.4 of Eisenbud's Commutative Algebra, there is the following comment. Most interesting Noetherian rings can be written as finitely generated modules over ...
0
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1answer
30 views

If $M$ is a monoid, is there accepted terminology for those elements $x \in M$ satisfying $xM = Mx$?

Suppose $M$ is a monoid and consider an element $x \in M$. Then we call $x$ central iff for all $m \in M$, it holds that $am=ma$. A vast weakening of this condition is to merely require $xM=Mx$. Lets ...
1
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1answer
15 views

ACC on chains of finitely generated submodules

Noetherian rings are those having ascending chain condition on ideals. There are also literatures concerning ACC on n-generated ideals (i.e., generated by n elements); see e.g, Commutative Rings with ...
1
vote
1answer
28 views

Show that the characteristic of $R$ is $2$ and that $R$ is commutative. [duplicate]

Let $R$ be a ring with identity such that $r^2=r$ for all $r\in R$. Show that the characteristic of $R$ is $2$ and that $R$ is commutative. My argument was the following. But, the thing I am not ...
1
vote
1answer
49 views

Decomposing polynomials over $\mathbb{Z}_3$

The first polynomial I had to decompose over $\mathbb{Z}_3$ is: $$x^2+x+1$$ I started by noticing that one root of it is $1$ so I thougth that I could factor this polynomial by $(x-1)$ but I ...
0
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2answers
79 views

What is the ideal? [closed]

Can you please explain to me why in $\mathbb{Z}/n\mathbb{Z}$, $n\mathbb{Z}$ is the ideal of $\mathbb{Z}$. Can you please explain with some examples. I'm having trouble with what is ideal? Appreciate ...
1
vote
1answer
52 views

Decomposing $x^4-5x^2+6$ over some fields

My book asks me to decompose $$x^4-5x^2+6$$ over: $K = \mathbb{Q},\\ K = \mathbb{Q[\sqrt{2}]},\\ K = \mathbb{R}$ For $K = \mathbb{Q}$, I substituted $x² = a$ to get: $$a²-5a+6 = (a-3)(a-2)$$ So ...
0
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1answer
43 views

Proving that the canonical ring homomorphism in $\mathbb{Z}[i] / \left< 5+3i \right>$ is surjective

Let $R=\mathbb{Z}[i]$ be the ring of Gaussian integers. Let $z=5 + 3i$ and let $I=\left< z \right>$. Let $\phi: \mathbb{Z} \rightarrow R/I$ be the canonical ring homomorphism. I am trying to ...
2
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1answer
23 views

Is my solution correct? (primes of the form $a^2+b^2d$ and their principal ideals)

I'm interested in the following exercise: A prime integer might be of the form $a^2 + b^2d$, with $a, b \in \mathbb{Z}$. Discuss carefully how this is related to the prime factorization of ...
2
votes
1answer
82 views

Free modules over commutative ring (possibly without unity) where free means having a LI spanning set

Let us define free module over a ring (possibly without unity) as: Def: M is said to be free module over ring R (possibly without unity) if there exist X subset of M such that X is LI and spans M. ...
3
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0answers
58 views

Prove $p(x) = x^3+x+1\in\mathbb{Z}_5[x]$ is irreducible over $\mathbb{Z}_5$ [duplicate]

We have: $$p(x) = x^3+x+1\in\mathbb{Z}_5[x]$$ First of all, I tried to prove that there are no roots in $\mathbb{Z}_5$, so I tried all of them: $$p(0) = 1\\p(1) = 3\\p(2) = 1\\p(3) = 1\\p(4) = 4$$ ...
4
votes
1answer
60 views

Find all $\mathbb{Z}_n$ in which $x^2+2$ divides $x^5-10x+12$

Find all $n\ge2$ such that $x^2+2$ divides $x^5-10x+12$ in $\mathbb{Z}_n$. To begin, I divided $x^5-10x+12$ by $x^2+2$ which gave me: $$x^5-10x+12 = (x^3-2x)(x^2+2)-6x+12$$ So, I guess I need to ...
1
vote
2answers
36 views

Cosets of $\mathbb{Z}_2[x]/<x²+1>$

This PDF says that the cosets of $$\frac{\mathbb{Z}_2[x]}{<x²+1>}$$ looks like the following: $$0+<x²+1>, x^{57}+x³+1+<x²+1>, (x²+1)+<x²+1>$$ Then it says that we can add ...
1
vote
2answers
72 views

Cosets of $\mathbb{R}[x] / \langle x²+1 \rangle$

I'm trying to understand this question and I've found this pdf which explains it a little better. My book has no intuition and don't even mention what it's doing, it just throw proofs at my face. So, ...
1
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2answers
18 views

If $R/J(R)$ is simple then $R$ is local

The question is what I said in the title :- If $R/J(R)$ is simple then $R$ is local. where $J(R)$ is Jacobson radical. I only have the idea about the converse ( If $R$ is is local then ...
0
votes
1answer
41 views

Every polynomial of odd degree $\ge 3$ over $\mathbb{R}[x]$ is reducible over $\mathbb{R}$

I need to prove that every polynomial of odd degree $\ge 3$ over $\mathbb{R}[x]$ is reducible over $\mathbb{R}$. If $p(x)$ is my polynomial, then I just have to prove that $p(x)$ has one real ...
1
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2answers
45 views

Show that $x^5-(3+i)x+2$ is irreducible in $(\mathbb{Z}[i])[x]$.

Show that $x^5-(3+i)x+2$ is irreducible in $(\mathbb{Z}[i])[x]$. $\mathbb{Z}[i]$ is a UFD and hence $(\mathbb{Z}[i])[x]$ is a UFD. So they are integral domains. Thus I can use Eisenstein's ...
2
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0answers
32 views

Extending ring metric to arbitrary module

I am working on my own on an idea I had and this one is a bit difficult so I am seeking a bit of inspiration and to see what already exists. Let $R$ be a given ring with a metric $d$ and $M$ being an ...
4
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2answers
159 views

Center of the Quaternions: Proof and Method

I have to calculate the center of the real quaternions, $\mathbb{H}$. So, I assumed two real quaternions, $q_n=a_n+b_ni+c_nj+d_nk$ and computed their products. I assume since we are dealing with ...
0
votes
1answer
47 views

The algebraic set $V$ is connected if and only if the coordinate ring $k[V]$ is not the direct sum of two nonzero ideals.

Let $V$ be an affine algebraic set in $\Bbb{A}^n$. Then $V$ is connected in the Zariski topology on $V$ if and only if $k[V] = k[\Bbb{A}^n]/I(V)$ is not the direct some of two ideals. I'm stuck ...
1
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2answers
29 views

Coprime elements in a PID satisfy that any of their powers are coprime

I have recently met this problem in my abstract algebra dealing with PID rings and coprimes stating: Let D be a PID ring $ a,b \in D $ two coprime elements. We are to show that for all $ m,n \in ...
1
vote
2answers
39 views

Why is this sequence of ideals an ascending chain? In proof of irreducible ideals are primary.

Let $P$ be an irreducible ideal in commutative ring $R$. Suppose $ab \in P$, $a \notin P$, and define $A_n = \{b^n x : x \in R\} \cap P$. Then "clearly $A_n \subset A_{n+1}$" says D&F. But ...
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0answers
15 views

central closure of semiprime rings

I have studied ring of quotients with 'Rings with generalized identities'.$R$:semiprime ring.$Q_{mr}$={$[f;J]| J$ is dense and $f:J\longrightarrow R$ right R-module homomorphism}$H$={$(f;J)| J$ is ...
3
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1answer
21 views

Is an injective map of $B$-modules also injective as an $A$-linear map if $B$ is an $A$-algebra?

I've been going through my submitted exercises again of my Commutative Algebra-class and I have the following question: Let $A$ be a commutative ring with unity. Given any injective homomorphism of ...
1
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0answers
27 views

Constructing a ring consisting of formal infinite series from a given ring

Let $A$ be an $\mathbb{N}$-graded $\Bbbk$-algebra, where $\Bbbk$ is a field, and where $\dim_\Bbbk A_n < \infty$ for all $n \in \mathbb{N}$. I can't see anything preventing me from constructing a ...
7
votes
2answers
131 views

Problem with the ring $R=\begin{bmatrix}\Bbb Z & 0\\ \Bbb Q &\Bbb Q\end{bmatrix}$ and its ideal $D=\begin{bmatrix}0&0\\ \Bbb Q & \Bbb Q\end{bmatrix}$

Let us consider the ring $ R:=\begin{bmatrix}\Bbb Z & 0\\ \Bbb Q & \Bbb Q\end{bmatrix} $ and its two-sided ideal $ D:=\begin{bmatrix}0 & 0\\ \Bbb Q & \Bbb Q\end{bmatrix} $. Let then ...
0
votes
2answers
37 views

Checking commutativity of a diagram of modules over some ring and what the commutativity of the diagram implies.

Suppose that you have the following diagram of modules over some ring: These are my questions: (1) To prove that the diagram is commutative, we needs to prove that $gf=kh$, $wf=rv$, $zh=uv$, ...
1
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0answers
20 views

How to recognize a monomial quiver algebra

Given a basic split finite dimensional algebra $A$ over a field K, A is isomorphic to $KQ/I_1$, for some quiver Q and a minimal(meaning it is generated by relations $x_i$, such that no relation is ...
0
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1answer
40 views

Elements of a ring which behave like multiplicative identity

Let $R$ be a commutative ring with unity. Let $0 \neq x \in R$, if whenever $xy = x$ for $1 \neq y \in R$ then ($y \in R$ is a zero divisor) there exists $0 \neq z \in R$ such that $zy = 0$. Example: ...
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0answers
16 views

Martindale quotient rings

Let R be prime ring.We know ($Q,+,.$) is ring under following operation: $(U,f)+(V,g)=(U+V,f+g)$$(U,f).(V,g)=(VU,fog)$ Iwant to find additive identity of $(Q,+)$ .So, I am looking any $(V,g)$ ...
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0answers
24 views

Finding all admissible Ideals of a given quiver with gap/qpa.

Let Q be a given finite quiver (with 1 point to make things easier for a start if necessary, for definitions see https://en.wikipedia.org/wiki/Quiver_%28mathematics%29 ) and fix a finite field K. Let ...
4
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0answers
39 views

Prove that in the ring of integers of a number field: a non-zero ideal coprime to $c\in\mathbb{Z_{\geq1}}$ has norm coprime to $c$

Let $\mathfrak{a}$ be a non-zero ideal of the RoI of a number field $A$, to say it is coprime to $c\in\mathbb{Z_{\geq1}}$ means: $\mathfrak{a}+cA=A$. The norm of $\mathfrak{a}$, or $N\mathfrak{a}$, ...
1
vote
1answer
38 views

Do the strongly vanishing elements of $R[[x]]$ form an ideal?

I've always been a bit annoyed by expressions like $$\sum_{n:\mathbb{N}} a_n$$ when the relevant limit doesn't converge, for the following reason: if you're not going to tell the reader what ring this ...
0
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0answers
18 views

Show that $[u,g]\in G'$ where $u\in N_U(G)$

Let $G=\langle H,g\rangle$ where H is an abelian subgroup of index $2$. Let $\Bbb{Z}G$ be the group ring and $u$ be a unit of $\Bbb{Z}G$ which normalizes $G$. Then we can write $u$ as ...
4
votes
1answer
63 views

Why are the quaternions not an algebra over the complex numbers?

I just began to study about algebras over rings and quickly came across the fact that the quaternions are not an algebra over the complex numbers. I would prefer an answer as elementary as possible.
0
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0answers
46 views

construction of free algebra

I can't understand the construction of a free algebra. I want to describe a free algebra. I write following from wikipedia. For $R$ a commutative ring, the free (associative, unital) algebra on $n$ ...
1
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0answers
43 views

Free $R$-module when $R$ is not unital

We can easily construct free $R$-module when $R$ is unital by setting $$R[S] = \{ f\colon S\to R\,|\, f\ \text{finitely supported}\}$$ and defining operations pointwise. The key here is that we can ...
2
votes
3answers
62 views

What are the units of Z[x]?

Where $\mathbb{Z}[x]$ is the ring of polynomials in $x$ with integer coefficients. The book I am studying says the unity of this ring is $f(x) = 1$ so then if some $p \in \mathbb{Z}[x]$ is a unit, ...
1
vote
1answer
17 views

order of $h_0 $ divides augmentation of $\alpha\in \Bbb{Z}H $

Let $H$ be an abelian group and $\Bbb{Z}H$ be its integral group ring. Now let $\alpha=\sum_{h\in H}a_h.h\in \Bbb{Z}H$ and $\alpha(1-h_0)=0$ for some $h_0\in H$. Why does this imply that order of ...
2
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2answers
34 views

Why does $\Bbb{Z}(G/H)$ has only trivial units?

Let $H$ be an abelian subgroup of index $2$ in $G$, and $G=\langle H,g \rangle$ then why is it only units of $\Bbb{Z}(G/H)$ are trivial (i.e. of the form $\pm\bar{g}$)? An arbitrary element of ...
0
votes
0answers
30 views

(Atiyah) If $A \subseteq B$ and $B \setminus A$ is multiplicatively closed then $A$ is integrally closed in $B$ [duplicate]

I've tried proof by contradiction, with $y \in B\setminus A$ and considering an integral expression $y^n = a_{n-1} y^{n-1} +\dots + a_0$ of least degree (hence $a_0 \neq 0).$ Then $a_{n-1} y^{n-1} ...
3
votes
3answers
58 views

Nonconstant polynomials do not generate maximal ideals in $\mathbb Z[x]$

Let $f$ be a nonconstant element of ring $\mathbb Z[x]$. Prove that $\langle f \rangle$ is not maximal in $\mathbb Z[x]$. Let us assume $\langle f \rangle$ is maximal. Then $\mathbb Z[x] / ...
0
votes
3answers
34 views

What rules does the identity element follow in a Ring.

I know that an element $e$ is called identity element in a group, say $(R, \cdot)$ if $e$ follows the rule $$ a\cdot e=e\cdot a=a $$ for all $a\in R$. My question is: What rules does $e$ have to ...
3
votes
1answer
61 views

Exercise from Kaplansky - Commutative Rings (1.1.3)

Exercise 3 in section 1-1: Let $P$ be a finitely generated prime ideal with annihilator 0. Prove that the annihilator of the module $P/P^2$ is $P$. (Hint: If $p_1,\cdots,p_n$ generate $P$ and $x$ ...
0
votes
1answer
38 views

$R$ semisimple artinian ring $\Rightarrow\varphi(R)$ is such [duplicate]

Let $R$ be a semisimple artinian ring, i.e. a right artinian ring with no nonzero nilpotents right ideals. We know that: $R\simeq M_{n_1}(D_1)\times\dots\times M_{n_t}(D_t)$ for ...
1
vote
1answer
28 views

Example for $(\mathfrak{a} + \mathfrak{b}) (\mathfrak{a} \cap \mathfrak{b}) \subsetneq \mathfrak{a} \mathfrak{b}$

In Atiyah-MacDonalds book on Commutative Algebra we have on page 6 the following statement ($\mathfrak{a}, \mathfrak{b}$ denote ideals in a ring): "(...) in $\mathbf{Z}$ we have $(\mathfrak{a} + ...
5
votes
1answer
80 views

Does a power-complete finite pasture exist?

Suppose we define a pasture to be an algebraic structure $\langle M, 0, +, \times, \wedge \rangle$ where $\langle M, 0, +, \times \rangle$ is a ring (not necessarily commutative or unital) $\wedge$ ...
5
votes
1answer
33 views

Example of an endomorphism on an abelian group that is not left multiplication

It is well-known that all endomorphisms on the abelian group ($\Bbb{Z}$,+) can be seen as a left multiplication by some element in some ring structure on ($\Bbb{Z}$,+); namely left multiplication by ...