Questions tagged [ring-theory]

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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Division algorithm in $\mathbb{Z}[i]$

Given the Gaussian integers $\alpha=5+3i,\beta=30+6i$ I want to find Gaussian integers $\gamma,\rho$ such that $\beta=\gamma\alpha+\rho$. Doing the division immediately we see that $\frac{\beta}{\...
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Formanek's proof of Eakin-Nagata theorem

I'm looking at the proof of Formanek of Eakin-Nagata theorem, that is also the proof given by wikipedia. The theorem says that if $A$ is a ring and $M$ a faithful finite $A$-module, such that ($\cdot$...
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Is this set a ring?

I am studying Ring Theory for the first time in my life- so the following question may be a very silly one. While trying to solve an (unrelated) exercise, this question clicked in me, and it has been ...
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1answer
80 views

Basic question on the category Cring

Let $A$ be a (commutative unitary) ring, let $I\subseteq A$ be a non-zero ideal. Are there any injective homomorphisms $A\to A/I$? I've been thinking for a while, keeping in mind that the homomrphisms ...
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1answer
42 views

Extension of ideals to powers series rings: example of $IR[[x]]\ne I[[x]]$.

It is not hard to show that if $I$ is a finitely generated ideal of a ring $R$ (here we assume $R$ contains identity and commutative), then $IR[[x]]$ is the ideal of all power series having their ...
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1answer
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If $R$ is Noetherian, show that any element of the form $x-a$, $a\in R$ is a nonzerodivisor in $R[[x]]$

Let $R$ be a commutative ring with identity, assume $R$ is Noetherian, show that any element of the form $x-a$, $a\in R$ is a nonzerodivisor in $R[[x]]$ I can prove this by flatness of completion. I ...
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61 views

Ideal of Polynomial Function on a Circle that Vanish at a Point

Let $R = \mathbb{R}[x,y]/(x^2+y^2-25)$ and $I$ the ideal of functions which vanish at the point $P = (3,4)$. I have proven that $I$ is generated by $(x-3,y-4)$ and that if $I= (f)$ for some $f \in R$, ...
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46 views

How to find units of rings?

In ring theory, I've been learning about units of rings, and although I'm able to understand the concept and write proofs involving it, I have a lot of struggle actually finding the units of specific ...
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1answer
24 views

proving that Every ring of prime order is commutative [duplicate]

I'm having some trouble with the following question: Let $R$ be a ring with order $p$, where $p$ is prime. Prove that $R$ is comutative. Because $(R,+)$ is a group then because of Lagrange's Theorem,...
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39 views

Splitting lemma, a detail

Let R be a commutative unitary ring and let $0\overset{}{\to}N \overset{f}{\to} M \overset{g}{\to} P \to 0$ a short exact sequence of R-modules. Now suppose that the sequence splits. If N and P are ...
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Determining all the unities of the ring $\mathbb Z [\omega]$

I'm having some trouble with the following question: Let $\omega$ be one of the two primitive cube roots of unity. Determine all the unities in the ring $\mathbb Z [\omega]$ The elements of $\mathbb ...
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2answers
65 views

Finding the Kernel and Image of $\mathbb Z \to \mathbb Z[i]/(1+3i)$, $x\mapsto x+(1+3i)$

I'm trying to apply the homomorphism theorem to the following function: $$h:\mathbb Z \to \mathbb Z[i]/(1+3i)$$ $$x\mapsto x+(1+3i)$$ Where $(1+3i)$ is the ideal generated by $1+3i$. I know that ...
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66 views

All maximal chains have the same length in $k[x_1,\dots,x_n]$

Let $k$ be a field, and let $A$ be a $k$-algebra of finite type that is an integral domain. Show that any maximal ascending chain of prime ideals in $A$ has length equal to $\operatorname {dim}A$. ...
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1answer
38 views

Clarify on Krull dimension and integrality

Let $f:A\to B$ be a homomorphism of rings, and let $I:=\operatorname{ker} f$. If $f$ is surjective, $\operatorname{dim}A\ge\operatorname{dim} B$. I'd say that this holds because a chain of prime ...
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Direct Proof that Artinian Rings have Stable Range 1

Is there a direct proof that any (right) Artinian ring has stable range 1? More precisely, let $R$ be a right Artinian ring and $a,b\in R$ be such that $aR+bR=R$. Can we prove that $(a-bt)R=R$ for ...
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3answers
89 views

Is it true that $\mathbb{Z}[\sqrt{79}]/(7, 1+\sqrt{79}) \cong \mathbb{Z}_7[X]/(\hat{1}+x)$?

I am trying to see whether $(7, 1+\sqrt{79})$ is a prime ideal in $\mathbb{Z}[\sqrt{79}]$. I tried doing this by looking at the factor ring $\mathbb{Z}[\sqrt{79}]/(7, 1+\sqrt{79})$. We have $\mathbb{Z}...
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1answer
20 views

How to show that with $P$ weakly prime ideal, if $\sqrt{0} \subsetneq P$, then $P$ prime

In "D. D. Anderson and Eric Smith, Weakly prime ideals, 2003". $R$ is commutative ring with identity There is a theorem: "Let $P$ be a weakly prime ideal of $R$. If $P$ is not prime, ...
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Resolution with Noether Normalization

Let $k$ be a field, and consider the $k$-algebra $A:=k[X,Y]/(XY)$. Give an explicit solution of this algebra using Noether normalization. So the task is to find a free $k$-algebra $k[t]$ such that $k[...
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If a,b are nilpotent elememts of a ring, then is (a*b) is nilpotent? [duplicate]

I had this in an exam and I cannot prove this as it seems that the ring must be conmutative, any hints?
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50 views

Argument with Noether's Normalization Lemma

Let $A$ be a $k$-algebra of finite type, where $k$ is a field, and suppose that $A$ is an integral domain. Noether's normalization states that exists a finite injection of $k$-algebras $\iota:k[x_1,\...
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1answer
25 views

Is every elements in weakly prime ideal is nilpoten element?

In "D. D. Anderson and Eric Smith, Weakly prime ideals, 2003". $R$ is commutative ring with identity There is a theorem: "Let $P$ be a weakly prime ideal of $R$. If $P$ is not prime, ...
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Exercise about integral closure

(i) Show that the integral closure of $\mathbb Z$ in $\mathbb Q(\sqrt{5})$ is $\mathbb Z[\frac 12(1+\sqrt{5})]$; it is known that this ring is a UFD. The fraction field of $\mathbb Z[\frac 12(1+\sqrt{...
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bitmasking polynomial $\mathbb{Z}_2[X]$ by multiplying with another one

Suppose I have a polynomial $\mathbb{Z}_2[X]$. That is, this polynomial represents a number in binary form (with just 1 and 0). Let's say that I want to extract the second bit by multiplying by ...
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1answer
35 views

Exercise with integral extensions

Let $R$ be an integral domain, and let $K$ be its field of fractions. Let $K\subseteq L$ a field extension. Is it true that, if $x\in L$ is integral over $R$, all the coefficients of the minimal ...
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1answer
26 views

Non-trivial zero divisors of Polynomial Quotient Ring

I'm writing below some things i found in an exercise: Let $f(x)=x^3+x^2+x+1 $ and $B:=\mathbb{Z}_2[x]/(x^3+x^2+x+1)$ Since the degree of $f(x)$ is $3$ and I found an evident root $f(1)=0$, then $f(x)$ ...
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43 views

Exercise 3.1.2, Bosch's Algebraic Geometry and Commutative Algebra

Let $R$ be a ring and $\Gamma$ a finite group of automorphisms of $R$. Show that $R$ is an integral extension of the fixed ring $R_{\Gamma}:= \{a \in R : \gamma(a) = a\ \forall \gamma\in\gamma\}$. It ...
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1answer
44 views

Ring of invariants preserves normality

Suppose we have a normal domain $R$ (i.e. integrally closed in its field of fractions) with a group $G$ acting on it by ring homomorphisms. I was wondering how one could prove that the ring of $G$-...
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Are euclidean rings the minimal condition for the greatest common divisor to exist?

In a textbook I read a proof that the greatest common divisor always exist and is unique for euclidean rings. Is there a more general algebraic definition for which the GCD always exists, or are ...
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42 views

Can the Chinese Remainder Theorem be proved for a commutative ring without unity? Or if $I+J \neq R$?

I apologize for this not being a more specific question. I am just wondering what conditions are "necessary" for the Chinese Remainder Theorem to hold true. I know that we need a surjective ...
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How called and what are the properties of rings in which there are elements such that division by them is sometimes zero? [closed]

Suppose there is a commutative ring in which there are "infinite" elements, of different infiniteness order, such that division of a finite element or infinite element of small order by an ...
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Find all ring homomorphisms $ ϕ: 2Z \rightarrow 3Z$ and $ ϕ: Z_{18} \rightarrow Z_{15}$ [duplicate]

Find all homomorphisms $ϕ$: $ ϕ: 2Z \rightarrow 3Z$ $ ϕ: Z_{18} \rightarrow Z_{15}$ My thoughts: $2Z = <2>$ Is a cyclic group. $ϕ(2) = 3k$ Since $ϕ$ Is a homomorphism: $6k = 2*3k = 2*ϕ(2) =...
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1answer
30 views

A question in the proof of Prime Avoidance Lemma

I was unable to prove Prime Avoidance Lemma for union of n prime ideals (did it for 2 prime ideals). So, looked on internet for help. I found a proof here in question where proof is written by OP but ...
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1answer
25 views

Question regarding a ring homomorphism of polynomials with coefficients in a field

I have a map $\sigma_a:K[x]\to K[x]$, where $K$ is a field, defined by $g(x)\mapsto g(x+a)$. I'd like to show that this ring homomorphism is in fact an isomorphism. I can easily see the map is ...
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2answers
87 views

If an algebra is a finite vector space it is a field

Let $A$ be an integral domain that is an algebra over a field $K$. Show that if $A$ is finite-dimensional as a $K$-vector space, it is a field. Is the converse true? Obviously the converse isn't true ...
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Find r(I) in following cases: a question related to inverse image and nilradical

This question is a part of the question:A formula for the radical of $\mathbb{Z}/n\mathbb{Z}$. The answerer of this question is suspended till December 2022 and I have a question in 2nd part of his ...
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1answer
68 views

$\operatorname{Frac}R/p\cong R_p/pR_p$

Let ${\mathfrak p}$ be a prime ideal of a ring $R$. Show that there is a canonical isomorphism $\operatorname{Frac}R/\mathfrak p\to R_{\mathfrak p}/{\mathfrak p}R_{\mathfrak p}$. Since this exercise ...
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24 views

Chinese Remainder Theorem (Lang proof) where is induction step used?

I have a hard time understanding where the induction step is used and why we even need induction by $n$ assuming we have ideals $I_1, \ldots, I_n$ and commutative ring with identity $A$. He shows it ...
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1answer
113 views

Is $1/2$ in $\mathbb{Z}[2^{1/2},2^{1/3},2^{1/4},...]$?

I was wondering if $\frac{1}{2}$ is in the ring $R=\mathbb{Z}[2^{1/2},2^{1/3},2^{1/4},...]$. I don't think it is, and I've been trying to prove by contradiction. So far I've shown that if this is true,...
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1answer
44 views

Proving that if $M$ is a finitely generated and projective $R$ module, then $M$ is also finitely presented.

Since $M$ is finitely generated $R$ module, we have an epimorphism $\pi : F \to M$, such that $F= R^{(n)}$ where $n$ is a positive integer. On the other hand, since $M$ is a projective $R$ module we ...
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1answer
47 views

Understanding $\Bbb Z_7[X] /(x^4+3x^2+x-3)$ [duplicate]

In $\Bbb Z_7[x]$ let $f(x)=x^4+3x^2+x-3$ and let $A = \Bbb Z_7[x]/(f)$ be the quotient ring. I'm asked to find the cardinality and the characteristic of $A$. I tried to rewrite and reduce the ...
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2answers
85 views

Exercise 1.1.4, Bosch's Algebraic Geometry and Commutative Algebra

Consider a ring $R$ and ideals $I_1,\dots , I_n$ such that $I_i+I_j=R$ for $i\neq j$. Show that the inclusion $\prod I_i\subseteq \bigcap I_i$ is an equality. I already proved Chinese remainder ...
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1answer
165 views
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Maximal ideals in finitely generated algebras over a field

Let $k$ be an algebraically closed field, let $A$ be a finitely generated $k$-algebra. If $m\subset A$ is a maximal ideal, show that it is generated by $\operatorname{dim}_k m/m^2$ elements. Consider ...
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1answer
87 views

If the associative law doesn't hold, can we define $a^n$? [duplicate]

I am reading "Higher Algebra" by A. Kurosh. The following sentence is in this book: Analogously, the associative law of addition leads to the concept of a multiple, $na$, of the element $a$ ...
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1answer
88 views

Proving if the ideal $\mathfrak{a}=(Y+2X^2,Z+3X^3,T^5-X^4-Y-Z)$ is prime or not in $K[X,Y,Z,T]$.

Note: I will abuse of notation and write $n$ instead of $n\cdot1$, for any $n \in \mathbf{Z}$. Obviously, $1$ being the identity of $K$. Applying the first isomorphism theorem and the correspondence ...
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12 views

Every prime/maximal ideal of ring of integers modulo n is <p> where p is a prime dividing n

Let R = $Z_n$ be the ring of integers modulo n such that n = $p_1^{a_1}.p_2^{a_2}...p_k^{a_k}$. I is a maximal/prime ideal of R $\iff \exists i\in $ {1,2,...,k} such that I = $<p_i>$
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45 views

Basic question about Zariski tangent space

Let $k$ be an algebraically closed field, and let $A$ be a Noetherian $k$-algebra. If $\Omega_{A/k}=0$, then $m/m^2=0$ for every maximal ideal $m\subset A$. Is the converse true? I would say yes, ...
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1answer
49 views

Homomorphism ring from integers to quotient ring of Gaussian integers

A very basic ring theory question, which I am not able to solve. Let $\phi: \mathbb{Z} \rightarrow \mathbb{Z}[i]/(2+3i) \text{ where } \phi(z) = z + (2+3i)\mathbb{Z}[i]$. Because I want to prove that $...
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1answer
141 views

Question about universal derivation $\Omega_{A/k}$

Let $k$ be a ring, let $A$ be a $k$-algebra. The universal derivation $\Omega_{A/k}$ is the (unique) $k$-module representing the functor of the $k$-derivations of $A$; suppose that $\Omega_{A/k}=0$. ...
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0answers
44 views

Element of a quotient group that equals to 0

I may be a bit confused but somehow I do not see why the following holds: Let $R$ be a ring and $I$ an ideal. Then we have the quotient $R/I$. Let $a+I \in R/I$. Why does it hold that if $a+I = 0 \...
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41 views

Proving characterisation of ideal being maximally disjoint from $1+\mathfrak a$

Let $R$ be a commutative ring and $\mathfrak a\triangleleft R$ a proper ideal of $R$. I wish to prove that $T^{-1}\mathfrak a$ is included in the Jacobson ideal of $R$, where $T=1+\mathfrak a$. In ...

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