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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-1
votes
1answer
33 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
58 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
21 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] $. ...
-1
votes
2answers
51 views

General questions about Polynomial Rings [closed]

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
27 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
2answers
31 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 ...
7
votes
1answer
118 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 ?
9
votes
1answer
93 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?
-4
votes
1answer
42 views

The mapping defines a unique automorphism [closed]

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 identity in $R$. ...
0
votes
1answer
82 views

Zerodivisor in $R[x]$. 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 ...
1
vote
2answers
28 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
52 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 ...
0
votes
2answers
29 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 $p(x)$ ...
2
votes
1answer
48 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
33 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
57 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 ...
1
vote
2answers
33 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
21 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 ...
1
vote
3answers
33 views

Annihilator of modules [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
33 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 ...
1
vote
1answer
39 views

Book recommendation on Primary decomposition of ideals [closed]

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 ...
-5
votes
1answer
43 views

In a ring $x^2= 0$ implies $x=0$. Then every idempotent is central. [closed]

In a ring $x^2= 0$ implies $x=0$. Then every idempotent is central.
0
votes
0answers
12 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
55 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 ...
2
votes
1answer
24 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 ...
0
votes
1answer
45 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 $Y$ ...
0
votes
0answers
20 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 ...
1
vote
1answer
58 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 ...
1
vote
1answer
34 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 ...
0
votes
1answer
29 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
74 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
31 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 $R$? ...
0
votes
1answer
43 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 ...
0
votes
1answer
39 views

Associated elements of a Euclidean domain have the same values of associated function [closed]

Let $D$ be a Euclidean domain and $d$ be the associated function. Show that if $a$ and $b$ are associates in $D$ then $d(a)=d(b)$. I'm not even getting how to begin this. Any hints are welcome. ...
1
vote
1answer
23 views

Sum of nil right ideals as an ideal

I have two questions: 1) If $S$ is the sum of all right nil ideals of a ring $R$ (with unity), is it true that $S$ is a two-sided ideal? It is clear for me that $S$ is a right ideal (and it is nil if ...
1
vote
1answer
35 views

An Artinian ring which is not Noetherian

Find an example of a ring which is Artinian but not Noetherian. Since a commutative Artinian ring is Noetherian , I know I have to look for a non- commutative Artinian ring. To start with a ...
1
vote
1answer
21 views

$N(R)$ when $R$ is a P.I. ring

The set of nilpotent elements $N(R)$ of a ring $R$ with identity is not necessarily a right ideal (or even a subgroup) as it is seen in the ring of $2×2$ matrices over $\mathbb Z$. But, my question ...
29
votes
0answers
313 views

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 $$x = a_0 + a_1 \alpha + \cdots + a_n \alpha^n$$ ...
1
vote
2answers
62 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. ...
0
votes
1answer
39 views

Commutativity of ring $R$ necessary for $\mathrm{Hom}_R(M,M')$ being an $R$-module

Why do we need $R$ to be commutative if we want $\mathrm{Hom}_R(M,M')$ (where $M$ and $M'$ are $R$-modules) to be an $R$-module itself? I tried to find out which axiom for modules does not hold if ...
0
votes
2answers
48 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 ...
0
votes
1answer
28 views

Basic quotient ring help

I am having a very hard understanding quotient rings. I never really understood quotient groups in abstract 1 but did well enough to get by. Now I am in abstract 2 and we hit quotient rings. I really ...
1
vote
0answers
74 views

Infinitely generated torsion free modules over PID

Let $R$ be a PID and $\mathbf{V}$ a torsion-free $R$-module, not necessarily finitely generated. If I understand it correctly, every rank 1 submodule of $\mathbf{V}$ is isomorphic to a submodule of ...
1
vote
0answers
15 views

units of the quotient ring of the integers over a prime power $[{\Bbb Z}/P^e\Bbb Z]^*$ is cyclic multiplicative group

I am studying Algebra as an extra curricular research project and in the reading I was assigned, the author somewhat offhandedly mentions that the units of ${\Bbb Z}/P^e\Bbb Z$, which is to say ...
0
votes
1answer
26 views

How to determine $(\mathbb{Z}[\alpha])^{\times}$

Let $\alpha = \sqrt{2p}$ with $p \equiv -3 \ [8]$ and we define : $\mathbb{Z}[\alpha]=\{a+b\alpha\ / a,b \in \mathbb{Z}\}$ $\mathbb{Q}[\alpha]=\{a+b\alpha\ / a,b \in \mathbb{Q}\}$ Let $N : ...
-1
votes
1answer
35 views

Quotient of a Euclidean domain

Let $A$ be a Euclidean domain such that its only invertible elements are $1$ and $-1$, and let $\varphi : A^* \to \mathbb{N}$ be a Euclidean function. Show that if $a \in A^*$ is a non-invertible ...
0
votes
1answer
22 views

Criterion of Eisenstein

Prove for every $n\in \mathbb{Z}_{\geq1}$ that the polynomial $X^n+Y^n-1$ is irreducible in $\mathbb{C}[X,Y]$ It is hinted that Eisenstein's criterion should be used for $Y-1\in \mathbb{C}[Y][X]$ ...
4
votes
1answer
55 views

Endomorphism Ring - Definition

Let $G$ be an Abelian group. We may consider the group $\big(\operatorname{End}(G), +\big)$. Next we may endow $\operatorname{End}(G)$ with the composition of functions to make it a ring. Anyway, it ...
1
vote
0answers
53 views

When a prime ideal is maximal differential ideal in a UFD?

Is the prime ideal $\langle X^{2}+Y^{2}-1\rangle$ a maximal differential ideal in differential ring $\mathbb{Q}[X,Y]$ with derivatives $D(X)=Y, D(Y)= -X$? I know there are maximal ideals like ...
1
vote
1answer
56 views

Isomorphism of Quotient ring $\Bbb Q[x]/\langle x^3\rangle$

$\Bbb Q[x]/\langle x^3\rangle$ is isomorphic to $R$. Find $R$. I know about $\Bbb Q[x]/\langle x\rangle$ is isomorphic to $\Bbb Q$ by the isomorphism $T(f(x))=f(x=0)$. $\Bbb ...