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
38 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
176 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
122 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 ...
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2answers
438 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 ...
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2answers
105 views

The number of $n\times n$ matrix over integer modulo $p$ field with determinant equal $1$

How to count the number of $n\times n$ matrix over integer modulo $p$ field with determinant equal $1$? I know that the number of invertible matrices is GL$(n,p)$. Have any ideas?
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1answer
53 views

Are there simple rings that are not division rings but have no zero divisors? [duplicate]

The only examples of simple rings I know that are not division rings a matrices over simple rings, but matrix rings always have zero divisors. Are there simple rings that are not division rings but ...
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1answer
144 views

Ring properties proofs.

Prove the following properties of a ring R: (a) For all a,b,c in R we have $a(b-c) = ab -ac$ (b) If R has multiplicative identity 1, then $(-1)(-1) = 1$. (c) If R has multiplicative identity 1, ...
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1answer
45 views

Isomorphism between two $K$-algebras

Consider a field extension $L\subseteq K$ and suppose that $I=(f_1,\ldots,f_m)$ is an ideal of $L[T_1,\ldots,T_n]$. Denote with $I^e\subseteq K[T_1,\ldots,T_n] $ the extended ideal of $I$ through the ...
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2answers
61 views

intersection multiplicity at non-zero point

Compute the intersection multiplicity of $f=x+y-2$ and $g=x^2+y^2-2$ at $(1,1)$. Is this the same as the intersection multiplicity of $f(x+1)$ and $g(x+1)$ at $(0,0)$ which I have computed to be 2? ...
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3answers
55 views

How to deduce whether $\sqrt{3}i \in \mathbb Q (u)$ or not, where u is a root of $x^3-2$

How can one deduce whether $\sqrt{3}i \in \mathbb Q (u)$ or not, where u is a root of $x^3-2$ ? $\mathbb Q (u) = \{c_0 +c_1u+c_2u^2 | c_i \in \mathbb Q\}$ Is it a trivial computation, or does this ...
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2answers
273 views

Computing the radical of an ideal

What is the best way to compute $\sqrt{(X^2-YZ,X(1-Z))}$ ? This is after using Nullstellensatz by the way as I thought it would be easier to compute a radical than finding the vanishing ideal.
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2answers
168 views

Computing irreducible components of algebraic set

Consider the algebraic set $V(X^2-YZ,X-XZ)$. Find the irreducible components of this set and show that $I(V)=(X^2-YZ,X-XZ)$. I reasoned that $X-XZ=0$ iff $X=0$ or $Z=1$. If $X=0$, we get $Y=0$ or ...
4
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1answer
63 views

Looking for a commutative ring satisfying certain conditions

I'm looking for a commutative ring $R$ (with unit) which is of characteristic 2 and which possesses elements $x$ and $y$ such that the following holds $x^2$ and $y^2$ are inverses of one another but ...
2
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1answer
57 views

Prove the integral property of $B[x] \cap B[y]$, where $y=x^{-1}$

Let $A$ and $B$ be two commutative rings with a unit element, with $B$ subring of $A$. Suppose $x$ is an invertible element in $A$ such that $xy=1$. Then prove that the intersection of the two rings ...
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1answer
34 views

Natural norm for the ring $\{a+b\sqrt{2}$ | $a,b \in \mathbb Z \} $?

I am working on showing that $\{a+b\sqrt{2}$ | $a,b \in \mathbb Z \} $ is an euclidean domain. There was a similar problem showing that $\{a+b\sqrt{-2}$ | $a,b \in \mathbb Z \} $ was an euclidean ...
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0answers
27 views

A problem involving ideals and prime ideals. [duplicate]

Please help me with a solution to this problem. Let $R$ be a commutative ring. Let $A_1, A_2$ be two ideals of $R$, and $P_1, P_2$ two prime ideals of $R$. Assume that $A_1 \cap A_2 \subseteq P_1 ...
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1answer
84 views

Associated Graded Module

I'm learning about Associated Graded Ring and Associated Graded Module. I got definition from Wiki: My questions: What's $I^nM$?, it's mean: $I^nM=am, a\in I, m\in M$? With above define, so: ...
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0answers
69 views

Any simple proof of that localization of a UFD is a UFD without using Kaplansky condition?

Using Kaplansky condition, the proof of this statement is quite easy. Before knowing this condition, I tried to prove this statement by actually showing the definition of UFD holds on $S^{-1}A$ ($A$ ...
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1answer
96 views

Proof of that entire $A$ is a UFD iff every nonzero prime ideal contains a nonzero prime element. (Kaplansky)

This is called Kaplansky condition: For $A$ entire, $A$ is a UFD iff every nonzero prime ideal contains a nonzero prime element. Here prime element means that $p \in A$ is prime iff $p$ is non-zero ...
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1answer
174 views

What is the difference between $\mathbb Z_n$ and $\mathbb Z / n \mathbb Z$?

I'm having trouble telling these apart. As far as I was aware $\mathbb Z /n \mathbb Z$ is the quotient ring where all multiples of $n$ will be zero - i.e. everything is done modulo $n$. But then what ...
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1answer
46 views

Proving the existence of unity in $R$, where $R$ is the ring of polynomials over complex numbers with $f(0)=0$.

My line of thought is this: we want to prove that there exists some $h(x)\in R$ such that $g(x)h(x)=g(x)$. Therefore $h(x)=1$ for all $x$. But if $h(x)$ is in $R$, then is it not equal to zero at ...
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1answer
236 views

Ring with non-trivial idempotent splitting as product of two rings

If a commutative ring $T$ has a non-trivial idempotent $e$, then it is easy to show that $f:T \rightarrow R\times S$ defined by $f(t)=(et,(1-e)t)$ is a ring isomorphism. My question is how to prove ...
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1answer
49 views

$\bar{\mathbb{Z}}\cap \mathbb{Q}\left[\sqrt{-3}\right] = \mathbb{Z}\left[\omega\right]$

How do you go about proving $\bar{\mathbb{Z}}\cap \mathbb{Q}\left[\sqrt{-3}\right] = \mathbb{Z}\left[\omega\right]$, where $\omega$ is $\frac{-1+\sqrt{-3}}{2}$? I have tried to approach it number ...
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2answers
65 views

Zero direct summand

I want a suggestion for the following: Let $M$ be a finitely generated $R$-module, where $R$ is a commutative Noetherian local ring with $1_R$, and $N$ a direct summand of $M$ such that $N⊆mM$, ...
2
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1answer
65 views

For $A$ integral domain, $S$ multiplicative subset, if $x/s \in S^{-1}A$ is irreducible, then $x$ is irreducible.

I need to prove the following: For $A$ integral domain, $S$ multiplicative subset, if $x/s \in S^{-1}A$ is irreducible, then $x$ is irreducible. First some observation: $x$ cannot be a unit nor $0$. ...
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0answers
75 views

Alternative proof for localization isomorphism

Let $f$ be an $A$-module morphism and $\operatorname{res}_{A_m}^A$ be the restriction of scalars functor from $A_m$-mod to $A$-mod. I'm curious if you have proven that for every maximal ideal ...
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1answer
97 views

Projective dimension zero or infinity

Let $R$ be a commutative Noetherian local ring (having unity) with the maximal ideal $m$, which consists only of zero-divisors. Then for any finitely generated $R$-module $M$, the projective ...
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1answer
71 views

Ultrafilter Lemma and Dimension Theorem

Reading on Wikipedia I find out that (the uniqueness in) the Dimension Theorem for arbitrary Vector Spaces can be proved using just the Ultrafilter Lemma (a strictly weaker version of Axiom of ...
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1answer
37 views

$\wedge^{2} \ (\mathbb{Q}/ \mathbb{Z}) = 0$

I have to prove that $$\wedge^{2} \ (\mathbb{Q}/ \mathbb{Z}) = 0$$ where $\wedge$ is the wedge product. Any hint ?
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2answers
62 views

Why is $\phi$ an epimorphism of rings?

$$\phi:\mathbb{Z} \rightarrow \mathbb{Z}_m$$ $$\phi(a)=[a]_m$$ Why is $\phi$ an epimorphism of rings ? A surjective function is when $$\forall y, \exists x : f(x)=y$$ Isn't it?? Is $\phi$ surjective ...
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3answers
42 views

Why are the rings $A$ and $B$ not isomorphic?

When the ring $A$ has no zero divisors, but the ring $B$ has, why are the rings $A$ and $B$ not isomorphic?? For example when $A=\mathbb{Z}[\sqrt{2}]$ and $B=\begin{Bmatrix} \begin{pmatrix} a & ...
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1answer
75 views

Is it possible to prove that this game is always winnable?

I was at my lunch table today and was trying to come up with a card game, and here is what I came up with: Let there be a standard deck of $52$ cards called $\mathbb{D}$, with four suits: spades, ...
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1answer
91 views

Literature to the ring $\mathbb{Z}[\phi]$ where $\phi=\frac{1+\sqrt{5}}{2}$ is the golden ratio

I know few about algebraic number theory but recently I stumbled upon the ring $\mathbb{Z}[\phi]$ where $\phi = \frac{1+\sqrt{5}}{2}$ is the golden ratio. It seems to be a very interesting object to ...
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1answer
59 views

Ideals Generated by an Element

I was wondering if anyone could give me a little explanation into ideals and principal ideals. I know that for $a \in R$, the principal ideal generated by $a$ is the set $$\langle a \rangle = ...
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1answer
69 views

Why is the order of a prime element well-defined?

This is in relation to the $p$-adic valuation on the field of fractions $F$ of an integral domain $D$. The idea is that for each $x \in F$ there is a unique maximal $k$ such that $x = p^k u v^{-1}$ ...
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2answers
26 views

For a commutative ring $A$, $S$ multiplicative subset, if $p \in A$ is prime, then $p/s$ is prime in $S^{-1}A$.

For a commutative ring $A$, $S$ multiplicative subset, if $p \in A$ is prime with $(p) \cap S = \emptyset$, then $p/s$ is prime in $S^{-1}A$. How do I prove this? Any suggestion or hint?
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0answers
118 views

Isomorphism between finite fields adjoining a root

Let $p(x)=x^3+x^2+1$ and $q(x)=x^3+x+1$ be polynomials over the field $\mathbb{Z}_2$. Let $\alpha$ be a root of $p(x)$ and $\beta$ be a root of $q(x)$. Now let $K=\mathbb{Z}_2(\alpha)$ and ...
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4answers
109 views

Is $\mathbb R[x] / \langle x^2+1 \rangle = \mathbb R[x] / \langle x^2+2 \rangle$?

I am starting to study rings. One of the first examples in my book about ring factors is $$\mathbb{R}[x] / \langle x^2+1 \rangle = \{ ax + b + \langle x^2 +1 \rangle \mid a,b \in \mathbb{R} \}$$ I ...
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2answers
580 views

When is nilradical not a prime ideal

Atiyah gives this criterion for nilradical to be a prime ideal.Nilradical is the intersection of prime ideals.Is nilradical prime iff there is only one prime ideal? ie Intersection of distinct prime ...
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2answers
235 views

Proving that Z(p) is a ring

Let $p$ be prime. Prove that $\mathbb{Z}(p)= \{a/b\ |\ a,b \text{ are elements of $\mathbb{Z}$ and $\gcd(b,p)=1$}\}$ is a ring. (This is called the ring of integers localized at $p$.) What should be ...
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1answer
33 views

Example of finitely generated module over valuation domain

I'm trying to find an example of finitely generated module over valuation domain with its generator. Is there anybody could help me to give the example beside the domain over itself?
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4answers
295 views

How can one visualize a homomorphic mapping.

It has been a year or so studying Group theory and Ring theory. Funnily enough, this is the part where i am able to solve most of the questions of the book quite easily, yet not fully understanding ...
6
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2answers
455 views

Exercise 2.27 Atiyah-Macdonald, absolute flatness

A commutative ring $R$ is absolutely flat if every $R$-module is flat. Prove that the following are equivalent: 1) $R$ is absolutely flat 2) Every principal ideal of $R$ is idempotent 3) Every ...
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1answer
44 views

Problem related to Cyclotomic Polynomials

I'm trying to show that if $p$ is prime, then $$x^{p-1}-x^{p-2}+x^{p-3}-...-x+1$$ is irreducible over $\mathbb{Q}$. I don't have an idea of how to start. I know the $p^{th}$ cyclotomic polynomial is ...
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2answers
97 views

Idempotent Elements of a Commutative Ring

I have to prove this statement and I'm a bit unsure how to go about it: Show that the set of all idempotent elements of a commutative ring is closed under multiplication. Furthermore, find all the ...
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3answers
505 views

Number of elements in the ring $\mathbb Z [i]/\langle 2+2i\rangle$

The question is : Show that $I=\langle 2+2 i\rangle$ is not a prime ideal of $\mathbb Z[i]$. Also find the number of elements in $\mathbb Z[i]/I$ and its characteristic. My try: I started with ...
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0answers
104 views

Resolution of module over polynomial ring

The problem is: Let $F$ be a field, and let $R = F[x_1, \ldots, x_r]$, the polynomial ring over $F$. Consider the $R$-module $M = R/(x_1, \ldots, x_r) \cong F$. Find a resolution of $M$ by free ...
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2answers
100 views

Make ring in natural way

Let $S$ be a subset in a commutative ring $R$, such that: $1 \in S$ $\forall x,y \in S \qquad xy\in S$ Define a relation $\sim$ on the Cartesian product $R\times S$ through ...
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0answers
23 views

For all ideals $I_1,I_2$, if $S^{-1}I_1 = S^{-1}I_2$ (localizations) then $I_1 = I_2$?

The wiki page says that the above implication holds if the ideals are prime. Here the multiplicative set $S$ contains $1$ but not $0$ and we are on a commutative ring $A$. What can we say about the ...
1
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1answer
55 views

Prove that $B$ is a subfield of $F$

If a subring $B$ of a field $F$ is closed with respect to multiplicative inverses, then $B$ is a field. Fields are commutative rings with unity, and every nonzero element has an inverse. A subring ...