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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Proofs with algebraic structures (rings)

If one is given a ring $R$ with a unity $u$, what are the steps one would have to take to prove that some element of $R$ named $s$ has a multiplicative inverse, where $-s$ also has a multiplicative ...
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Trace of nilpotent matrix over a ring

Let $R$ be a commutative ring with unity, and $n$ a positive integer. Let $A\in \mathfrak{M}_n(R)$ such that there exists $m\in \mathbb N$, for which $A^m=0$. Is it true that there exists $\ell\in ...
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Center of a ring projective?

If $R$ is a ring and $Z(R)$ denotes $R$'s centre, then when is $R$ projective as an $Z(R)$-module?
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Global dimension of the center

Let $R$ be a ring. Must the global dimension of the centre $Z(R)$ of the ring $R$ always be atmost that of $R$ itself? I mean is it generally true that: $D(Z(R)) \leq D(R)$ (where D is the global ...
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Ideal of direct sum of rings

If $R$ is a ring and $S_1, S_2$ are subring of $R$ such that $R=S_1\oplus S_2$, Is there any relation between ideal of $R$ and Ideal of $S_1,S_2$? In particular I mean under which condition we can ...
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Prove that any subfield of $\Bbb R$ contains $\Bbb Q$

Prove that any subfield of $\Bbb R$ must contain $\Bbb Q$. Now for any subfield $F$ of $\Bbb R$, $1\in F$ so, $\Bbb Z \subset F \Rightarrow \Bbb Q \subseteq F$. Have I done it correctly?
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Why most discussion on ring(like module) is over PID, not UFD? [closed]

I think many properties and discussion are based on unique factorization. Like On UFD, irreducible element is also prime. So why these analysises are not based on UFD?
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Socle of submodule relative to the module

in these notes i am reading i am told that the socle of $K$ (where $K \subset M$ , and $M$ is a module) is = $K \cap$ Soc $ M$ But why is this? i see the intuition but cannot formalize a proof ...
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Prove $-1$ and $1$ are the only units in $\mathbb{Z}$ [closed]

Prove $\mathbb Z^*=\{-1,1\}.$ I have a proof, which is posted as an answer below. I'm looking for an alternate proof.
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Homomorphisms of semi-simple submodules

I managed to show $\forall$ R-Module M, $\exists$ unique semi-simple submodule $sM \subset M$ containing every semi-simple submodule of M, by showing that the direct sum of semi-simple submodules are ...
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Rings and Semi-simple rings

I'm failing to see which of the following are semi-simple rings, any help would be appreciated. $\mathbb{C}[X]$, the group ring $\mathbb{Q[Z]}$ and $\begin{pmatrix} \mathbb{Z} & \mathbb{Q}\\ 0 ...
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Is $r_1 \cdot f_1 + r_2 \cdot f_2 $ uniformly distributed?

Consider $f_1$ and $f_2$ are fixed polynomials, $r_1$ is a random linear polynomial, $r_2$ is a random polynomials, degree($r_2$)=degree($f_i$)=$d$. We define $f_i$ and $r_i$ over $R[x]$ where $R$ can ...
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Could you give an example of not every finitely generated Z[x]-module a direct sum of cyclic modules?

Could you give an example of not every finitely generated Z[x]-module a direct sum of cyclic modules? I have no idea about the example, could you give me some ideas? Thank you.
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When considering the commutative ring $R$ over itself, if its submodule is free, then the submodule equals to the module?

When considering the commutative ring $R$ over itself, then this $R$-module is isomorphic to $R$, but if $I$ is an ideal of $R$, then it is a submodule, if this submodule is free too, then it is ...
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Show that field of fraction of a commutative domain is an indecomposable module which is not finitely generated

I came across this problem and get stuck for quite sometime. Problem: Let $R$ be a commutative domain that is not a field. Let $F$ be its field of fractions. Show that $F$ is an indecomposable ...
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Direct summand of modules proof

I'm have problems trying to show this, any help would be appreciated. Show $i: N \subset M$ is a direct summand iff $\exists$ a module map $r: M \to N$ s.t $ri = 1_{N}$, and that any complement of ...
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To prove given $ r \cdot f_1+f_2\cdot (s+1)$ one who knows $f_2$ cannot find out what $f_1$ is

We define the polynomials $r,f_1,f_2,s\in R[x]$. Where $r$ is a random degree 1 polynomial and $s$ is a random polynomial such that: $\deg(s)=\deg(f_1)=\deg(f_2)$. Let $R$ be $\mathbb {Z}_q$ where $q$ ...
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Showing that the operations defined on a right ring of fractions are well-defined

My problem comes from Goodearl & Warfield's "An Introduction to Noncommutative Noetherian Rings". Let X be a right Ore set of regular elements in a ring R. Define a relation $\sim$ on $R ...
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Writing a particular polynomial as product of irreducibles in various rings.

I want to factor the polynomial $x^3-10x+4$ into a product of irreducibles over each of the fields $\mathbb{Z}[i]$,$\mathbb{Q}[\sqrt{2}]$, $\mathbb{Q}[\sqrt{2},\sqrt[3]{2}]$, $\mathbb{Z}/ 11 ...
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Finding all ideals in $\mathbb{Q}[x]/I$, where $I$ is the ideal generated by $p(x)=x^2(x^2+x+1)$

I want to find all ideals in $\mathbb{Q}[x]/I$, where $I$ is the ideal generated by $p(x)=x^2(x^2+x+1)$. I know that $$\mathbb{Q}[x]/I \cong \mathbb{Q}[x]/(x^2) \times \mathbb{Q}[x]/(x^2+x+1)$$ I ...
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Is $R$ PID if every submodule of a free $R$-module is free?

Let $R$ be a commutative ring. Before I proved that every submodule of a free $R$-module is free over a P.I.D. Now I'm trying to prove the reciprocal, if every submodule of a free $R$-module is ...
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Question regarding a ring homomorphism.

I have the following questions. I define a homomorphism $\phi:R\to R\times \mathbb{Z}$ by $\phi(r)=(r,0)$. I can't understand the following: 1.) My notes describe that this homomorphism is not ...
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Maximal ideal in $K[x_1,…,x_n]$ [duplicate]

I'm having some difficulty with this homework problem: If $A=K[x_1,...,x_n]$, $K$ a field and $a_1,a_2,...,a_n $ $\in K$. The ideal $m=<x_1-a_1,...,x_n-a_n>$ is maximal.
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Open and closed sets for j-Spec $A$.

The following is from Matsumura, Theorem 4.10 Let $A$ be a ring and $M$ a finite $A$-module. (i) For any non-negative integer $r$ set $$U_r = \{p \in \text{Spec} \space A | M_\mathfrak{p} ...
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Injective ring homomorphism on $A/I\cap J \rightarrow A/I\times A/J$

Suppose $A$ is a ring with proper ideals $I$ and $J$. I have showed the function $\theta:A\rightarrow A/I\times A/J$ given by $\theta(x) = (x+I,x+J)$ is a ring homomorphism with kernel $\ker \theta = ...
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Show that $a,b$ are units in a ring

Let $R$ be a ring with unity and assume that $R$ has no nonzero zero-divisors. Let $a,b\in R$, and assume that $ab=1$. Show that $ba=1$, and therefore $a,b$ are units. I think this question boils ...
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If $R$ is isomorphic to a ring $S$, show that $S$ has this same property: $n*1_{S}=0$

I am having some trouble with a homework problem... Hoping someone may be able to provide some insight and a bump in the right direction! Suppose $R$ is a commutative ring with unity $1_{R}$, and has ...
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Let k be a field and let p, q ∈ N be two prime numbers such that p · 1 = q · 1 = 0. Show that p = q.

My current train of though is letting p =/= q then proving that q must be divisible by p, the contradiction then being that q is prime. But I'm not sure how to go about doing this.
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How many unique combinations of sets can we get?

Starting with $x$, which is a positive integer or zero, and $y$ a second positive integer or zero, with $y \ne x$, we can create lists. Set $p$ a prime greater than 2, $\alpha = (p-1)/2$, and ...
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Show that the Frobenius homomophism is bijective.

Let k be a finite field with characteristic $p ≥ 2$, show that the homomorphism $F:k\rightarrow k$ where $x \mapsto x^p$ is bijective. Could someone please explain the statement $p ≥ 2$ to me, as ...
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Is $x^2-2$ irreducible over R and Q?

I'm not sure if it is valid to say that $x^2 - 2$ can be factorised to $2\cdot\left(\frac 12x^2 - 1\right)$ for it to be reducible in Q. Though I know $(x + \sqrt{2})(x - \sqrt{2})$ works in the ...
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Show that $x^3 + 3x^2 + 9x + 3$ and $x^3 + 3x^2 + 3x − 4$ are irreducible in $\mathbb{Z}[x]$

I need to show, as stated in the title, that $x^3 + 3x^2 + 9x + 3$ and $x^3 + 3x^2 + 3x − 4$ are irreducible in $\mathbb{Z}[x]$ I know that in case of second polynomial, if $f(x-1) = x^3 - 5$ which ...
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Rings and prime ideals

While doing some exercises about rings and prime ideals i got stuck with the following: Having a ring R: {$a + b \sqrt7 | a,b \in \mathbb{Z}$}, being a subring of $\mathbb{R}$, and knowing that ...
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Ring homomorphism and units homomorphism

I am trying to show that given $\phi: R\rightarrow R'$ is a ring homomorphism $R, R'$ are rings, there exist a homomorphism between the group of units that is $\phi': R^\times\rightarrow R'^\times$ is ...
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1answer
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Irreducibility over a field of rational functions

Let $K$ be a field, and let $f,g\in K[Y]$ be coprime with $\deg(fg)\geq1$. How do I prove that $f-gX\in K(X)[Y]$ is irreducible? I tried the "generic" approach of assuming the existence of a ...
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Proving $(\mathbb{C},\mathbb{C})$ Is Not A Field [duplicate]

Let's $(\mathbb{C},\mathbb{C})$ be a ordered paired of elements form $\mathbb{C}$ when $\mathbb{C}$ is defined as (a,b). addition and multiplication is defined as in $\mathbb{C}$. How do I prove it ...
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Why an ideal in a ring is a submodule of a free module over that ring

I know an ideal in a ring is a module over this ring, but I don't know why it's a submodule of a free module, what's the free module? Thank you.
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1answer
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Is every finite ring a matrix algebra over a commutative ring?

In this MO answer it is stated that every finite ring is a direct sum of finite-dimensional algebras over $\mathbb{Z}/p^k$ for varying $p$ and $k.$ What I am wondering is the following: Can every ...
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Proving that the Gaussian integers have no zero divisors

The Gaussian integers are the ones of the form $m + ni$ where $m,n$ are both integers. I need to show that given any two Gaussian integers $a$ and $b$, $ab = 0$ must imply that $a = 0$ or $b = 0$. I ...
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Why is the naive notion of a product ideal not necessarily additively closed? [duplicate]

Considering the product ideal $IJ = \{ \sum_{i=1}^n a_ib_i | a_i \in I, b_i \in J \forall i\}$, I've always seen it written that the more naive notion $IJ = \{ ij | i \in I, j \in J\}$ is not an ideal ...
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What is $\mathbb{F}_7[X]$?

I do not understand what sets like these are. I know what something like $\mathbb{Z}_7$ is. It is the ring of integers modulo 7 so it is equal to ${0,1,2,3,4,5,6}$. But what is $\mathbb{F}_7[X]$ equal ...
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First Isomorphism Theorem for Rings

I am trying to get a deeper understanding of the First Isomorphism Theorem, but am having trouble seeing the "natural mapping" that my textbook says exists. Upon looking it up more online I came ...
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Verification of an R-Module isomorphism between $R^n$ and its dual

With one step at a time, I am getting slightly more used to $R$-Modules. Let $R$ denote a commutative Ring with $\mathbb{1}$ and $n$ a natural number. For the tuple $a:= (a_i)_{i=1}^n \in R^n$ we ...
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1answer
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Determining if (3) is a maximal ideal in $\mathbb{Z}[\sqrt{7}]$.

As far as I can tell, the tools I have for determining if an ideal I of a ring R is maximal is either: Determine another ideal it is contained within, or look at the quotient ring $R/I$ and determine ...
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Left ideals of matrix rings are direct sum of column spaces?

Let $\mathbb K$ be a field and $M_n(\mathbb K)$ be the ring of the $n\times n$ matrices with entries in $\mathbb K$. Let $C_j\subset M_n(\mathbb K)$ be the subspace of all matrices which have all ...
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Quotient by the ideal $I=(2\sqrt{2})$ in the ring $R=\mathbb Z[\sqrt{2}]$

Let $R=\mathbb Z[\sqrt{2}]$ and $I=(2\sqrt{2})$ be the principal ideal generated by $2\sqrt{2}$. Let $a,c\in \{0,1,2,3\}$ and $b,d \in \{0,1\}$ and suppose that $$a+b\sqrt{2}+1=c+d\sqrt{2}+I$$ ...
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Show that $a$ is irreducible iff $au$ is irreducible where $u$ is invertible

$R$ is integral domain. Show that $a$ is irreducible iff $au$ is irreducible where $u\in R^*$. My Try: Lets assume $a$ is irreducible and $au = bc \implies a=bcu^{-1}$. We know that $u\in R^*$ so ...
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$R$ integral domain : $u\in R^*, a \text{ is prime} \iff au \text{ is prime}$

$R$ integral domain : $u\in R^*,\; a \text{ is prime} \iff au \text{ is prime}$ I started by looking at $auu^{-1}$. What should I do next? I'd be glad for help. Note: $u \in R^*$ meaning is $u$ ...
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Proving that an ideal is prime - is it correct?

I need to prove that although $X^2 + 3X +1 \in \mathbb{Z} [X]$ is irreducible, the ideals $(5,X^2 + 3X +1 )$ and $(11, X^2 + 3X +1)$ are not prime. I know that an ideal $I$ is prime iff ...
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$R/\mathfrak p$ not always a UFD [closed]

I am looking for a nice counterexample that for a UFD $R$ and $\mathfrak p\subset R$ a prime ideal, $R/\mathfrak p$ is not always a UFD as well.