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
76 views

Do polynomials make sense over non-commutative rings?

One could think of polynomials rings as sort of a derived ring (a ring of functions $f: \mathbb{N}^m \mapsto R$ such that $f^{-1}(R \setminus \{ 0 \} )$ is finite), but from what I can tell, we are ...
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2answers
99 views

For a ring if we have: $\forall a\in R$ with $za=az=z$ does that mean $z=0$

For a ring if we have: $\forall a\in R$ with $za=az=z$ does that mean $z=0$? If we have $x+z=x$ for all $x\in R$ ($x+z=z+x$ as it is a ring) then we show z is unique and call it 0. For 0 it is ...
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1answer
4k views

How to prove that the inverse of a matrix is unique?

The ring of matrix is not an integral domain. How to prove that the inverse is unique?
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2answers
118 views

Sum of unit and nilpotent element in a noncommutative ring.

Something similar is asked here but it is not exactly the same thing. An element $a$ of a ring $R$ is called nilpotent if $a^n=0$ for an $n \in \mathbb{N}$. Show that if $u$ is a unit and $a$ is ...
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1answer
65 views

Symmetric powers of ideal quotients in a local ring.

Let $R$ be a local ring and $I \subset R$ any ideal. When is it the case that $(I \: \backslash I^2)^n = I^n \: \backslash I^{n+1}$? Put another way, when is the natural map $\text{Sym}^n(I/I^2) ...
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1answer
518 views

History of the terms “prime” and “irreducible” in Ring Theory.

In ring theory, a nonzero, nonunit element $p$ of a integral domain is called irreducible if $p=ab$ implies that exactly one of $a$ and $b$ is a unit, and it's called prime if $p\mid ab$ implies that ...
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1answer
70 views

Soft Question: Geometry of Rings

So.. I was thinking about something last night and literally have no idea where to start. Can the ring axioms be connected to topology? If so, how? Basically, the motivation for my question has to ...
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3answers
83 views

$A/m^n$ is Artinian for all $n\geq 0$ if $A$ is a Noetherian ring and $m$ maximal ideal.

How to prove : $A/m^n$ is Artinian for all $n\geq 0$ if $A$ is a Noetherian ring and $m$ maximal ideal. Any suggestions ?
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2answers
107 views

$1+\sqrt{2}$ is a unit in $\mathbb{Q}[\sqrt{2}]$. True or False

I believe that the statement is True, and this is my argument: Since there exists an element $((1-\sqrt{2})/(1-(\sqrt{2})^2)\in\mathbb{Q}[\sqrt{2}]$ such that their products gives $1$ (multiplicative ...
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1answer
82 views

Retraction for rings?

For abelian groups, the existence of left inverse or right inverse of a homomorphism can be characterized by looking at whether the image or kernel splits the group. Is there an analogous ...
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2answers
246 views

$\mathbb{Z} _{29}$ is a field. True or False.

My answer was True and this is my argument: Since $\mathbb{Z}_{n}$ has got $2$ operations plus the other properties of a ring, I figured that it is indeed a ring. On the other hand, since ...
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0answers
19 views

Is $\min \deg$ in a dirichlet subring interesting or is it always $1$?

Let $s \in C$. Let $D = A[[n^{-X}]]$ be a subring of the formal (or absolutely converging on a region; whatever is needed) Dirichlet series with base ring $A$. Define a minimal Dirichlet series for ...
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1answer
53 views

Comparing injective dimensions in a short exact sequence

If $0→A→B→C→0$ is an exact sequence in the category of $R$-modules ($R$ commutative having unity) with injective dimensions of $A$ and $C$ both $≤n$, is that of $B$ also $≤n$? It seems to me that ...
3
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1answer
64 views

Give an example of a ring in which there exist $x,y,z \in R\setminus\{1,0\}$ such that $x$ is a unit, $y$ is nilpotent, and $z$ is idempotent.

Give an example of a ring in which there exist $x,y,z \in R\setminus\{1,0\}$ such that $x$ is a unit, $y$ is nilpotent, and $z$ is idempotent. I have an idea of the form $R = \mathbb{Z}/ ...
2
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1answer
113 views

Suppose a finite ring $R$. Show that each $x \in R$ is exactly one of a unit, nilpotent, or $x^k$ is idempotent

Suppose a finite ring $R$. Show that each $x \in R$ is exactly one of a unit, nilpotent, or $x^k$ is idempotent. I know I must show this in cases. Case 1: Suppose $x$ is a unit. Then there ...
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1answer
255 views

Matsumura, Exercise 18.8: Cohen-Macaulay and (not) Gorenstein [duplicate]

I need an answer to the exercise 18.8 of Matsumura's book:" Commutative ring theory", and generate an algorithm if possible. Let $k$ be a field and $t$ an indeterminate. Consider the subring $A = ...
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1answer
107 views

If $R$ is a noetherian local ring, then every 2-generated ideal has finite projective dimension iff $R$ is a UFD

This question is about zcn's comment on the answer to this question. It's a good point. So I ask it for use of everybody: if $R$ is a noetherian local ring, then every 2-generated ideal has ...
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1answer
98 views

Readings for Noether

I'm studying the theory of Noether but I have only 4 pages of lecture notes with no details or examples. Are there any good lecture notes or chapters you know about? In my lectures the basics of ...
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1answer
144 views

How to calculate $\operatorname{Spec} \mathbb{C}[x,y]/(y^2-x^3)$

Is there a general method for calculating things like $\operatorname{Spec} \mathbb{C}[x,y]/I$ ? Maximal ideals are $ \{(x-\tilde{a},y-\tilde{b}): b^2-a^3=0\}$ because of ...
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1answer
147 views

Projective dimension of all principal ideals is finite. Is R an integral domain? [closed]

$R$ is a noetherian ring in which projective dimension of all principal ideals is finite. Is $R$ an integral domain? What condition can be added on it to be a regular ring? thanks for any help. ...
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2answers
85 views

Algebra - Gaussian integers

Let $\mathbb{Z}[i]=\{ a+bi : a,b \in \mathbb{Z}\}$ be the ring of Gaussian integers. Let $x,y \in \mathbb{Z}[i]$ with $y \neq 0$. Show that there exist $q,r \in \mathbb{Z}[i]$ such that $x = yq + r$ ...
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3answers
35 views

Each $c \in R^*$ divides each polynomial of $R[X]$

We consider that $R$ is a commutative ring with $1_R$. Each $c \in R^*$(if we see it as a constant polynomial), divides each polynomial of $R[X]$. ($c \in R^*$ means that $c$ is invertible.) I ...
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1answer
35 views

A cyclic right ideal which is not finitely generated

I am looking for a two-sided ideal $I$ in a ring with identity such $I$ is not finitely generated as a left ideal but it is cyclic as a right ideal.
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1answer
72 views

Principal ideals as generated groups

This seems like a pretty simplistic question, but I can't find a solid, non-ambiguous answer to it. The question I'm given: Is $I$ a principal ideal of $R$? Given: $R=\mathbb{Z}$ and $I=\left\langle ...
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2answers
120 views

Is it a field or not?

Let $S \subset R$, $R$ ring. Is $S$ a field, knowing that $\displaystyle{R=M_2(\mathbb{R}), \text{and } S= \begin{Bmatrix} \bigl(\begin{smallmatrix} 0&0 \\ 0&a \end{smallmatrix}\bigr): a ...
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0answers
44 views

Brauer Group - A measure of complexity?

I have seen many authors state that the Brauer Group in some way measures the complexity of a field. I've convinced myself that the Brauer group of the reals is Z/2Z, and that the Brauer group of an ...
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0answers
75 views

Equations in the semiring of f.g. modules

Let $R$ be a commutative ring. Then we may consider the semiring $G(R)$ of isomorphism classes of finitely generated $R$-modules with $+ = $ direct sum, $* = $ tensor product, $0 = $ zero module, $1 = ...
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158 views

$x \otimes y - y \otimes x \neq 0$ in $I \otimes_{R} I$

Let $R = k[x,y]$ , $I = (x,y)$ , $k$ is a field. I want to prove that : 1) $x \otimes y - y \otimes x \neq 0 $ in $I \otimes_{R} I$ 2) $x \otimes y - y \otimes x $ is a torsion element My ...
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1answer
201 views

Is every commutative simple ring a field?

I'm aware that every commutative simple ring with unity is a field. Is this true even for rings for which we don't know if they have a unity?
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1answer
172 views

Properties of $Hom_{\mathbb{Z}}(\mathbb{Q}, \mathbb{Q} )$

I have to prove that : 1) $Hom_{\mathbb{Z}}(\mathbb{Q}, \mathbb{Q}) \cong \mathbb{Q}$ as abelian groups 2) $End_{\mathbb{Z}}(\mathbb{Q}) \cong \mathbb{Q}$ as rings What I have done: 1) We can ...
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0answers
35 views

Proof about central simple algebras

I'm going through the following proof (part 2) from "Central Simple Algebras and the Brauer Group" by Eduardo Tengan. I understand down to the point where the author says "moreover, applying the ...
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3answers
233 views

If $a,b$ is an $R$-sequence, then $ax-b$ is prime (Eisenbud, Exercise 10.4)

This is the exercise mentioned above: Let $a,b$ be regular sequence over a domain $R$. Prove that $ax-b$ is a prime of $R[x]$. Thank you for your answer!
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0answers
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“Filters” of associative rings

Does there exist some good notion of "filters" for arbitrary associative rings with unity that would generalize filters of Boolean rings? For me, if $R$ is an associative ring with unity $1$, it ...
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1answer
222 views

Maximal Ideals of Matrix Ring

Let $D$ be a division algebra. I'm trying to show that all simple $M_n(D)$ modules are isomorphic to $D^n$. I know that $M_n(D)=D^n\oplus D_n \oplus \dots \oplus D^n$ (n terms). I have that if $S$ is ...
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1answer
101 views

Exercise about Krull Dimension/Eisenbud, Exercise 10.1

I can't solve this exercise. If someone can help me, thanks a lot. Let $R$ be a Noetherian ring, and $x$ an indeterminate. Prove that $\dim R[x,x^{-1}]=\dim R+1$. Thank for your answers!
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1answer
596 views

Addition and Multiplication table for Ring/Ideal

I'm not sure if it's possible to show it here, but how would the addition and multiplication table look like for R/I (where R is rings with ideal I) when $$ R = Z_{12} \text{ and } I = \{0,3,6,9\} ...
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3answers
276 views

A field is a nonzero commutative ring …

I was confused when I read this statement. I thought a Ring must have the additive inverse $0$. Does the statement imply that there is no zero in a field?
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1answer
94 views

Units in $\mathbb{Z}[\sqrt{d}]$

I'm proving that if $d \in \mathbb{Z}, d < -1$ and $d$ is square-free , then the only units of $\mathbb{Z}[\sqrt{d}]$ are $\pm1$. I proved it, but I never used that $d$ is square-free. Where ...
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1answer
81 views

Kronecker's Theorem and showing isomorphism

The question: Let $K = \mathbb{Q}$ and $f = X^4 + 1$. Show that $f$ is irreducible in $\mathbb{Q} [X]$. Explain why $\mathbb{Q} [X]/⟨X^4 + 1⟩ ≃ \mathbb{Q} [(1 + i)/\sqrt{2}]$. I have shown that $f$ ...
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1answer
57 views

Proving that these isomorphisms are the $only$ isomorphisms

How would I prove that the only isomorphisms $\theta :\mathbb{Z[i]} \rightarrow \mathbb{Z[i]}$ are the identity map and $\theta(a+bi) = a-bi$? I have no idea how to start this, my first thought was ...
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1answer
60 views

Finding some homogeneous generators of an ideal.

Suppose that $\mathfrak a$ is an homogeneous ideal of $K[T_1,\ldots, T_n]$ where $K$ is a field of characteristic $0$ and $T_1,\ldots,T_n$ are indeterminates. Moreover suppose that $\mathfrak a$ has a ...
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3answers
292 views

$\mathbb Z_{mn}$ isomorphic to $\mathbb Z_m\times\mathbb Z_n$ whenever $m$ and $n$ are coprime

How to show that $\mathbb Z_{mn}$ is isomorphic to $\mathbb Z_m \times\mathbb Z_n$ when $m$ and $n$ are coprime? It is easy to show that the natural map from $\mathbb Z_{mn}$ to $\mathbb Z_m ...
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1answer
67 views

Equivalent condition for Jacobson radical

Matsumura, Commutative Ring Theory, page 3, asks this: If $x \in A$ has the property that $1 + Ax$ consists entirely of units, then $x \in \operatorname{rad}(A)$. Prove this. How to show this?
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2answers
45 views

Existence of Composite Ring Homomorphism

I am at a loss as to proving the following result: Let $\phi:R \to S$ be a ring homomorphism with $R$ commutative and $I$ an ideal of $R$ such that $I \subseteq ker \phi$ . Then there exists a ring ...
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2answers
61 views

Greatest common divisor of polynomials over $\mathbb{Q}$

I have two polynomials: $f: x^3 + 2x^2 - 2x -1$ and $g: x^3 - 4x^2 + x + 2$. I have to do two things: find $gcd(f,g)$ and find polynomials $a,b$ such as: $gcd(f,g) = a \cdot f + b \cdot v$. I have ...
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0answers
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Change of basis - free submodule

Let $R$ be a commutative ring and $(e_1, \dotsc, e_n)$ be a basis for $R^n$, the free $R$-module of rank n. Let $A$ be an $n \times n$ matrix with entries in $R$. Let $f_i = \sum \limits_{j=1}^n ...
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1answer
41 views

Determining the form of elements in quotient ring of multivarible polynomial ring

I want to find the elements in the quotient ring ${F_2}\left[ {x,y} \right]/\left( {xy + 1} \right)$. I think that the elements of this ring are of the form ${a_0} + {a_1}{x^n} +{a_2} {y^m}$, ${a_i} ...
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3answers
217 views

how to compute product in the given ring

$(-3,5)(2,-4)$ in $\mathbb{Z}_4 \times \mathbb{Z}_{11}$ I get the answer as $(2,3)$. The answer given in the solution is $(2,2)$. Can someone explain how?
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1answer
140 views

A submodule of a free module over a PID

$R$ is a principal ideal domain and $F$ is a free module over $R$ of infinite rank with basis $\{e_1,...,e_n,...\}$. Is it true that the $R$-submodule of $F$ spanned by $\{e_1,...,e_n\}$ has a ...
4
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2answers
562 views

Is the center of a ring an ideal?

Let $Z(R) = \{ a \in R : ax = xa,\text{ for all $x \in R$}\}$ Is $Z(R)$ an ideal of $R$? Attempt: I already proved that $Z(R)$ is a subring of $R$. I would say yes, since if $x \in R$, then $xa$ is ...