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
42 views

Significance of $I$ being irreducible in $R/I$

Let $f(x) = x^2 + 1$ and let $g(x) = x^2$ Now $f$ is irreducible in $\mathbb{Z_3[x]}$ while $g$ is reducible in $\mathbb{Z_3[x]}$ Let $R = \mathbb{Z_3[x]}/(f(x))$ Let $S = \mathbb{Z_3[x]}/(g(x))$ ...
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2answers
465 views

Determine all monic irreducible polynomials of degree $4$ in $\mathbb{Z_2[x]}$ [duplicate]

Determine all monic irreducible polynomials of degree $4$ in $\mathbb{Z_2[x]}$ Well these polynomials will be of the form - $a_0 + a_1x + a_2x^2 + a_3x^3 + x^4$ So we have four coefficients that ...
4
votes
0answers
69 views

Morita theory for simplicial rings

My question is the following: is there an analog of Morita theorem in the simplicial setting? I mean, we can define two simplicial rings $A,B$ to be simplicially Morita equivalent is the categories ...
0
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1answer
112 views

Determining the maximal ideals in $\mathbb{Z_2[x]}/(x^3 + x + 1)$

Let $R = \mathbb{Z_2[x]}/(x^3 + x + 1)$ The elements of $R$ will be all polynomials in $\mathbb{Z_2[x]}$ with degree less than $x^3 + 2x + 1$. So the elements of $R$ are - $\overline{0}, ...
0
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0answers
94 views

Show a polynomial is irreducible, ring homomorphisms, prime numbers

Let $p$ be a prime number. 1) Show that the linear transformation: $r_p:\mathbb Z[x] \to \mathbb F_p[x]$ that replaces the coefficients with the remainders of their division by $p$ is a homomorphism ...
1
vote
1answer
122 views

Semisimple ring problem

Prove that: $R$ is a semisimple ring $\Longleftrightarrow$ Every right $R$-module is injective (projective) My try: $R$ is semisimple ring $\Longleftrightarrow$ Every right $R$-module is semisimple ...
1
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1answer
54 views

Any submodule U of V such that the module V/U is completely reducible must contain the radical?

Want to prove: Given an $R$-Module $V$ and $rad(V)$ which I define to be the intersection of all maximal submodules of $V$. I want to show that if, for some submodule $U$ of $V$, we have $V/U$ ...
2
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1answer
448 views

A module as an external direct product of the kernel and image of a function

If $f:A\rightarrow A$ is an R-module homomorphism such that $ff=f$, show that $$A=Ker\,\,f\oplus Im\,\, f$$ Here is a part of what I made as a proof. Let $a\in A$. $f(a)\in Im\,\,f$, ...
1
vote
1answer
122 views

Determing all monic irreducible quadratic polynomials in $\mathbb Z_2[x]$

The monic quadratic polynomials in $\mathbb Z_2[x]$ are - $x^2, x^2 + 1, x^2 + x, x^2 + x + 1$ $x^2 = x \cdot x$ so is reducible $x^2 + 1 = (x + 1) \cdot (x + 1)$ so is reducible $x^2 + x = x \cdot ...
2
votes
2answers
143 views

Procedure for showing a polynomial is irreducible in $\mathbb{Q[x]}$

Let $f(x) = x^5 − 6x^2 + 21x + 13$ What is the procedure for showing $f(x)$ is irreducible in $\mathbb{Q}[x]$?
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1answer
41 views

Basic question on irreducibility

Let $f(x) = x^2 + x + 1$ in $\mathbb{Z[x]}$ $f$ does not have a root hence it is irreducible. So the only factorisations of $f(x)$ are ones that contain units, i.e. $1 \cdot (x^2 + x + 1)$ and ...
2
votes
1answer
39 views

Show that $\sum_{n=0}^\infty (order\ {S_n})q^n=\prod_{m\ge 1}(1-q^m)^{-1}$

Let $T=\mathbb (C^*)^2$ acts on $\mathbb C[x,y]$ via $(t_1,t_2)(x,y)=(t_1x,t_2y)$, let $S_n$ be the set of ideals $I$ of $\mathbb C[x,y]$ such that $TI=I$ and $\mathbb C[x,y]/I$ is $n$-dimensional ...
0
votes
1answer
72 views

Linearly independent subsets over extension of scalars

Let $k$ be a field, $D_{1},D_{2}$ division rings containing $k$ and $V$ a $k$-vector space of finite dimension and assume also that $V$ is a $D_{1}-D_{2}$ bimodule. Suppose $W$ is a $k$ basis of ...
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votes
2answers
161 views

Polynomial ring and the free algebra

In the Algebra book of Mac Lane there is an exercise in Chap. IV which tells me to construct a polynomial ring $A[X]$ for any set (not necessarily finite) $X$ ($A$ a ring), and to give correct the ...
3
votes
1answer
115 views

Let $K$ a field with characteristic $p>0$. Show that $\{x \in K : x^{p^n} =x \}$ is a subfield.

Let $K$ a field with characteristic $p>0$. I've shown that for every positive $n$ the set $\{ x^{p^n} : x \in K \}$ is a subfield of $K$, I did this by showing that $F:K\to K: x \mapsto x^{p^n}$ is ...
3
votes
3answers
62 views

Do the real numbers form a division ring with operations $a\oplus b=a+b+\frac12$ and $a\odot b=a+b+2ab$?

Is $(\Bbb R,\oplus,\odot)$ a division ring, where $$a\oplus b = a+b+\frac12$$ and $$a\odot b = a+b+ 2ab?$$ I have only issues with $\odot$. It doesn't work for inverse of $$a=\frac{-1}{2}$$ ...
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3answers
155 views

Determing all maximal ideals in $\mathbb Z / 4\mathbb Z[x]/(x^4 + 2x + 1)$

Say I have a quotient ring $\mathbb Z/4\mathbb{Z}[x]/(x^4 + 2x + 1)$. What is the process for finding the maximal ideas of this ring?
2
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1answer
47 views

In $\mathbb{Z_2[x]}/(x^2 + x +1)$, what is $\overline{x} \cdot \overline{x}$?

In $\mathbb{Z_2[x]}/(x^2 + x +1)$, what is $\overline{x} \cdot \overline{x}$? $\overline{x} \cdot \overline{x} = \overline{x^2}$ If I divide $\overline{x^2}$ by $(x^2 + x +1)$ I get an answer of $1$ ...
1
vote
1answer
50 views

Ideals of $\mathbb Z[x]$ containing $(3, x^3 - 1)$.

I would like to diagram the complete lattice of ideals of $R = \mathbb Z[x]$ containing the ideal $I = (3, x^3 - 1)$. By the lattice isomorphism theorem, each ideal of $R$ containing $I$ corresponds ...
0
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0answers
51 views

Questions on surjective homomorphism from $\mathbb{Z_3[x]} \rightarrow \frac{\mathbb{Z_3[x]}}{(x^3 + x + 1)}$?

Say I define a homomorphism $$\lambda : \mathbb{Z_3[x]} \rightarrow \mathbb{Z_3[x]}/(x^3 + x + 1)$$ such that $$\lambda(z) = \overline{z}$$ Then if I take the element $x^5 + 2x^2 +1$ in ...
4
votes
1answer
136 views

What are the elements of the quotient ring $\mathbb{Z_3}[x]/(x^3 + x^2)$?

$$R = \mathbb{Z_3}[x]/(x^3 + x^2).$$ As $\mathbb{Z_3}$ is a field we have that every polynomial in $\mathbb{Z_3}[x]/(x^3 + x^2)$ of degree less than ${x^3 + x^2}$ is a distinct element in $R$. So I ...
8
votes
1answer
566 views

Do the non-units in a commutative ring form an ideal?

Do the non-units in a commutative ring form an ideal? The following are my thoughts on this. Have I made any incorrect assumptions? Let $R$ be a commutative ring. Let $a, b \in N$ with $N$ being the ...
2
votes
2answers
43 views

Under what conditions is this true? A finitely generated ideal minus a generator equals an ideal containing that.

Let $(r_1, \dots, r_{s-1}) \subset I \subset J = (r_1, \dots, r_s)$ be ideals in a commutative ring $R$. In other words $r_s \notin I$. Also, let $\{r_1, \dots, r_s\}$ be a minimal generating set ...
5
votes
1answer
95 views

Prime ideals in $C[0,1]$ and ultrafilters

I'm looking for prime ideals in the ring $C[0,1]$ of continuous functions $[0,1]\longrightarrow \mathbb{R}$. I raised that question recently and got good answers but now I'd like to improve a bit the ...
2
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2answers
335 views

Find a non-principal ideal (if one exists) in $\mathbb Z[x]$ and $\mathbb Q[x,y]$

I know that $\mathbb Z$ is not a field so this doesn't rule out non-principal ideals. I don't know how to find them though besides with guessing, which could take forever. As for $\mathbb Q[x,y]$ I ...
0
votes
2answers
199 views

Show that a polynomial $f(x)$ over a field $k$ is irreducible if and only if the polynomial $f(x+1)$ is irreducible

I have literally no idea how to do this. I tried writing out $f(x+1)$ and rewriting it but didn't seem to get anywhere. I suspect I should use the degree of the polynomials but don't know how.
0
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0answers
82 views

Dimension of local ring as vector space over $\mathbb C$

I want to know what the dimension of each of the local ring $\mathbb C[x,y]_p/(y^2-x^7,y^5-x^3)$ is, where $p\neq (0,0)$ over $\mathbb C$-vector space. I know the dimension of it in the origin point, ...
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3answers
268 views

what is the relationship between vector spaces and rings?

Can you show me an example to show how vector and scalar multiplication works with rings would be really helpful.
1
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1answer
88 views

problem involving polynomial ring over a field

So far, I have the following: Could someone show me why $h \mid x^q -x$ and also why $h$ has a root $b$ in $F$? I can figure out the rest. Thank you for your help!!
0
votes
1answer
33 views

Number of elements in $\{ x^{p^n}: x \in K \} \subseteq K$ for a field $K$

I have another question about ring theory. Let $K$ b a field, with $p>0$ its characteristic. Define the set $T \:= \ \{ x \in K: x^{p^n}=x \} $. I had to prove that $|T| \leq p^n$. Research ...
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2answers
73 views

In a ring if a product is invertible with one factor is not a divisor of zero then both factors are invertible

Let $(A,+,.)$ be a ring, and $a$ et $b$ two elements of $A$ such that $ab$ is invertible and $b$ is not a divisor of zero. I want to show that $a$ and $b$ are invertible. My try: since $ab$ is ...
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0answers
76 views

How to establish this as being true or false?

Let $R$ and $R^\prime$ be rings such that $R$ has unity $1$ and $R^\prime$ has no $0$ divisors; let $\phi \colon R \to R^\prime$ be a homomorphism such that $\phi [ R ] \neq \{ 0^\prime\}$, where ...
7
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0answers
226 views

Homomorphic Compression

Can there be an algorithm such that, given plaintext data P,Q, and compression function e, Such that if we treat P and Q as a number (a series of bits): $$\begin{eqnarray*}e(P + Q)& =& e(P) ...
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4answers
108 views

Consider the ideal $\langle 1+i \rangle$ in $\mathbb{Z}[i]$.

Consider the ideal $\langle 1+i \rangle$ in $\mathbb{Z}[i]$. $(a)$ Make use of the given description of this ideal, $\hspace{75pt}$ $\langle 1+i \rangle = \{a+bi:a+b \text{ is even}\}=\{\alpha\in ...
3
votes
3answers
257 views

Isomorphisms and the Fundamental Homomorphism Theorem

Let $$ R=\left\{ \begin{bmatrix} a & b \\ 0 & a \end{bmatrix} : a,b∈ℝ\right\}⊂M_2(ℝ) $$ and $$ I=\left\{ \begin{bmatrix} 0 & b \\ 0 & 0 \end{bmatrix}: b∈ℝ\right\}. $$ Identify the ...
1
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1answer
59 views

Isomorphism- product of ideals

If $A$ and $B$ are two rings,and $\alpha$ is an ideal of A and $\beta$ is an ideal of B, then $\alpha \times \beta $ is an ideal of $A \times B$. I have to prove that $A \times B / \alpha \times ...
0
votes
1answer
49 views

Abstract Algebra (Ring Homomorphisms and Ideals)

Show that the equation $y^2=4$ has at least $4$ solutions in the ring $\mathbb{Z}/5[x]/\langle x^2+1\rangle$. What do you conclude? My main question about this is what $\mathbb{Z}/5[x]/\langle ...
2
votes
2answers
78 views

A Ring of Square Roots

$\forall x \exists y(x = y \cdot y)$ is true for the trivial ring and $\mathbb{Z} /2 \mathbb{Z}$. Is it true in any other $\mathbb{Z} /n \mathbb{Z}$ rings?
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2answers
95 views

Tensor product of a ring with itself

If $R$ is a commutative ring then $R \otimes_{R} R \cong R$. Is this still true if $R$ is non-commutative?
1
vote
1answer
129 views

Krull dimension of this local ring

I want to know what the Krull dimension of this ring $\mathbb C[x,y]_p/(y^2-x^7,y^5-x^3)$ is, where $p\neq (0,0)$. I know the dimension of it in the origin point, but I don't know other cases.
0
votes
1answer
135 views

What is the Krull dimension of this local ring

I want to know what is the dimension of this ring $\mathbb C[x,y]_{(0,0)}/(y^2-x^7,y^5-x^3)$. I don't know how to do that. If I suppose $y^2=x^7$ I will get a higher degree of $x$.
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2answers
88 views

Question on decomposing a noetherian ring into product of PIDs

Let $R$ be a reduced noetherian rings of dimension $d$, $S$ be a multiplicative set of all regular elements of $R$, and $K=S^{-1}R$ be localisation of $R$. Show that $K[T]$ (polynomial in 1 ...
0
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1answer
35 views

Decide the dimension of maximal ideals

Let $A=\mathbb C[x,y]/(y^2-x^3,y^5-x^3)$. I want to know the dimension of each maximal ideal over $\mathbb C$. Actually I can't decide it's maximal ideal. And how to decide its dimension?
2
votes
1answer
393 views

Ring of continuous functions on $\mathbb{R}$, maximal ideal, quotient

Let $I(S) = \{f \in \mathcal{C}(\mathbb{R}) \ | \ \ \forall x \in S: f(x)=0\}$ I've already proven that it is an ideal in the ring $\mathcal{C}(\mathbb{R})$. However, I have troubles proving that ...
0
votes
1answer
78 views

The maximal ideals in the polynomial ring

Let $A=\Bbb C[x,y]/(xy,y(y-a))$. I want to know the maximal ideals in $A$. I don't know how do deal with it, I'm confused by the structure of $A$.
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0answers
141 views

First Order Definitions of Finite

I would like some predicates in the language of first order Peano arithemetic (PA) that are true for the standard natural numbers and false for other types of numbers like negative numbers, fractions, ...
2
votes
0answers
68 views

homework: rings, matrices and polynomials

$A,B$ are both $n \times n$ and diagonal matrices. Prove that there is a matrix $X$ which is $n \times n$, and polynomials $p$ and $q$ such that $A= p(X), B= q(X)$ Is this true for ANY 2 matrices (we ...
0
votes
1answer
86 views

homework about homomorphisms : find all the homomorphisms

Find all the continuous homomorphisms $T:\mathbb{R} \rightarrow \mathbb{R}$ Find all the homomorphisms $T:\mathbb{C} \rightarrow \mathbb{C}$ (complex field) such that $T(x)=x$ for every $x$. ...
2
votes
1answer
86 views

Simple ring and field

We know that the center of a simple ring with unity is a field. But I couldn't make an example of a ring which is not simple but its center is a field. Is it possible? Please give a hint.
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0answers
164 views

Subring with maximal ideals (prime avoidance). Proof verifying and small question

Let $t∈\Bbb N$ and let $p_1, \dots ,p_t$ be $t$ distinct prime numbers. Show that $$R = \{α∈\Bbb Q : α = m/n \mbox{ for some } m ∈ \Bbb Z \mbox{ and } n∈\Bbb N \mbox{ such that } n \mbox{ is ...