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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Which of the following form an ideal in this ring?

Let $C(R)$ denote the ring of all continuous real-valued functions on with the operations of pointwise addition and pointwise multiplication. Which of the following form an ideal in this ring? The ...
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1answer
85 views

Is there a name for an algebraic structure like this?

I'm self studying abstract algebra. I see that in rings there's no requirement for a multiplicative inverse. Is there something similar except with no requirement for an additive inverse. For ...
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1answer
38 views

example for $(\mathfrak{a}:\mathfrak{b})^e\subsetneq (\mathfrak{a}^e:\mathfrak{b}^e)$?

showing that the inclusion holds is an easy exercise, but can someone give an example where the inclusion $(\mathfrak{a}:\mathfrak{b})^e\subseteq (\mathfrak{a}^e:\mathfrak{b}^e)$ is strict?
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1answer
79 views

About direct sum and direct product of integral algebras

Is an easy exercise to prove the following assert: Let $B_0, \dots , B_n$ integral algebras over $A$. Then $\bigoplus_{i=0}^n B_i$ is integral over $A$. Is this true if the sum is infinite? What ...
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3answers
353 views

Example of a commutative ring that is Artinian but not Noetherian

I want to give an example of a commutative ring that is Artinian but not Noetherian. Is there any examples not very difficult? I considered the ring $\mathbb{Z}+x\mathbb{Q}[x]$. It is not ...
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1answer
150 views

counterexample that $M$ is not finitely generated $R$-module and $M$ has no maximal submodule

In the proof of Nakayama Lemma, the following proposition is used: let $R$ be a commutative ring with identity and $M$ be a non-zero finitely generated $R$-module. Then $M$ has a maximal submodule. ...
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1answer
77 views

Is $\cos x$ irreducible as a power series?

Let $\mathbb{Q}_{\mathrm{ent}}[[x]]$ be the ring of entire functions with rational coefficients. Is $$ \cos x \;=\; \sum_{n=0}^\infty (-1)^n\!\frac{x^{2n}}{(2n)!} $$ irreducible in ...
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1answer
138 views

Ideal, not finitely generated

This exercise is driving me insane. I think there might be a mistake in it. Consider the ring $R$ of matrices of the form: $ R = \lbrace \begin{pmatrix} z & q_{1} \\ 0 & q_{2} \\ ...
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65 views

A problem in division rings and Brauer group

Suppose that $D, E$ are two division rings in the Brauer group of $F$ ($Br(F)$), where $F$ is local field. Show that $D\otimes_FE$ is a division ring iff $([D:F],[E:F])=1$.
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184 views

Are the ideals of any integral domain $R$ trivial only?

Are the ideals of any integral domain $R$ trivial only?
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1answer
96 views

Question concerning Eisenbud's theorem on matrix factorisations

I have the following question: Let $S$ be a commutative regular local ring and $\mathfrak{n}$ be its maximal ideal. Let $f\in\mathfrak{n}$ be a non zero-divisor in $S$ and let $m\geq 1$ ne a natural ...
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1answer
175 views

How should I understand $R[x]/(f)$ for a ring $R$?

The following is a proposition in Artin's Algebra: Proposition 11.5.5 Let $R$ be a ring, and let $f(x)$ be a monic polynomial of positive degree $n$ with coeeficients in $R$. Let $R[\alpha]$ ...
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5answers
121 views

Is the notation $X²$ an abuse of language in a polynomial ring.

I wonder if the notation $$ P(X) = a_{0} + a_{1}X + a_{2}X²$$ where $X$ is an indeterminate variable is an abuse of notation. Is $X^2$ just $X_2$? Put it another way, what's the meaning of the power ...
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1answer
44 views

Are there any ordering for the elements of an ideal?

In page 17 of "Introduction to Grobner basis" by R. Froberg, the multiplication of two ideals is defined as follows: $ \mathcal{a} \cdot \mathcal{b} = \left\{ \sum_{i=1}^{k} a_i \cdot b_i \vert ...
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362 views

Is $\sqrt{2}\in{\Bbb Z}[\sqrt{2}+\sqrt{3}]$ true or not?

Motivated by the positive answer to the following question: Is $\mathbf{Q}(\sqrt{2}, \sqrt{3}) = \mathbf{Q}(\sqrt{2}+\sqrt{3})$? I'm curious about whether ${\Bbb ...
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3answers
136 views

proper ideals in the principal ideal domain

I'm to prove that every proper ideal is a product of maximal ideals which are uniquely determined up to order. I have no idea even how to start in the proof to solve this question :( May anybody help ...
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2answers
204 views

Polynomial in two variables with zero constant coefficient form principal ideal?

Let $F$ be a field, and $F[x,y]$ the ring of polynomials in $x,y$. Let $J$ be the subset of all polynomials $P(x,y)$ in $F[x,y]$ such that $P(0,0)=0$. Then $J$ is an ideal. Is $J$ a principal ideal?
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484 views

Proof that $\mathbb{Z}_p$ is an Integral Domain iff $p$ is prime.

The proposition given in my lecture notes is: $\mathbb Z_p$ is an integral domain iff $p$ is prime. The first part of the proof is written as follows: Suppose the integer n is not a prime and ...
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1answer
204 views

localization in algebraic geometry

It is often asserted in commutative algebra texts that localization is important in algebraic geometry. I would appreciate some precise examples which show the utility of the concept in this context. ...
2
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1answer
241 views

Unique factorization domain and principal ideals .

If R was a unique factorization domain, can we deduce that for a nonzero element d in R, d has a finite number of divisors? I need this in solving this question " If R is a unique factorization ...
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3answers
111 views

unique factorization domain

I'm asked to solve If $R$ is a unique factorization domain and for $a$, $b$ two elements that are relatively prime in $R$ and $a$ divides $bc$, then $a$ divides $c$. While trying to prove this, ...
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1answer
59 views

Show that a given ring is a field with four elements

Let $R = ( \mathbb{Z} / 2 \mathbb{Z} ) [t]$ be the ring of polynomials with coefficients $\mathbb{Z} / 2 \mathbb{Z}$, $f = f(t) = t^2 + t +1$, and $g = t^2 +1$. Show that: (1) $R/(f)$ is a field with ...
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1answer
112 views

Maximal ideal not containg the set of powers of an element is prime

In the midst of attempting to prove that for a commutative ring $A$ with identity, and an ideal $I$ of $A$, $I = rad(I)$, where $rad(I) = \{x: x^m \epsilon I, m >0\}$, implies that $I$ is an ...
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1answer
147 views

Infinite direct product of rings free.

Let $A$ be a commutative ring (viewed as an $A$-module over itself) that is not a field. Are there some conditions that guarantee that $\prod_{k=0}^\infty A$ is free? What if $A=\mathbf{Z}$ or more ...
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316 views

Questions about a commutative ring with exactly three ideals

Let $R$ be a commutative ring with identity. Assume that $R$ has exactly three distinct ideals: $\{0\},I, R.$ 1) Show that if $a \in R-I$, then $a$ is a unit in $R$. 2) Let $a,b\ne0$ in ...
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3answers
34 views

Divisibility of polynomials in $\mathbb{Z}_n[x]$

For what values of $n$ is $x^2+1$ a factor of $x^5+5x+6$ in $\mathbb{Z}_n[x]$? I know how to divide in $\mathbb{Z}[x]$ (with long division), but what should I do here with $\mathbb{Z}_n[x]$, and it's ...
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1answer
195 views

if $f \in A[x]$ is a zero divisor, then there exists $a ≠ 0$ in $A$ such that $af = 0$. [duplicate]

The title of the question indicates what I am attempting to prove, that if $f$ is a member of a polynomial ring over a commutative ring with identity, and $f$ is a zero divisor, then there exists a ...
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1answer
78 views

Polynomial division in $\mathbb{Z}[x]$

I see an exercise that says Find the quotient and remainder when $x^3+2$ is divided by $2x^2+3x+4$ in $\mathbb{Z}[x]$. Since $\mathbb{Z}$ is not a field, I cannot do ...
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2answers
97 views

Is $(\mathbb{Z}/p\mathbb{Z})[x]$ an integral domain?

Is $(\mathbb{Z}/p\mathbb{Z})[x]$ an integral domain? Take two polynomials $a_nx^n+\ldots+a_1x+a_0$ and $b_mx^m+\ldots+b_1x+b_0$, and suppose their product is $0$. Then we have that either $a_n=0$ or ...
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1answer
267 views

Infinite ring with nonzero characteristic [duplicate]

I was wondering as I read about characteristic of a ring: Is there an infinite ring with nonzero characteristic? We have $1+1+\ldots+1=0$, but that doesn't seem to imply that the number of elements in ...
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2answers
58 views

Divide with remainders in a ring

How is it works ? What is different between divide with remainders in a ring and without ? e.g I have this question: Calculate $\frac{6x^5+2x^4+5x^3+x+2}{5x^3+x^2+6}$ in the ring ...
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Definition of primary ideal question

A primary ideal (in a commutative ring with unity) is an ideal $J$ for which if $ab\in J$, then either $a\in J$ or $b^n\in J$ for some integer $n\geq 1$. So it also implies (due to commutativity) that ...
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1answer
76 views

Containment between prime ideals and maximal ideals

In a commutative ring (with unity), is it true that (a) any maximal ideal is a prime ideal? (b) any prime ideal is a maximal ideal? (b) is almost certainly false, because a maximal ideal is a ...
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1answer
93 views

How general are $\mathrm{Aut}$ groups and $\mathrm{End}$ rings?

For any locally small category $X$ and object $A\in X$ the set $\mathrm{Aut}_X(A)$ is a group w.r.t. composition $\circ$. For any locally small abelian category $X$ and object $A\in X$ the set ...
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1answer
171 views

Rings that are isomorphic to the endomorphism ring of their additive group.

Every ring is isomorphic to a subring of the endomorphism ring of it's underlying group. That's Cayley's theorem for rings. What can we say about rings that are isomorphic to the endomorphism ring of ...
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2answers
131 views

when is a ring a free module over a subring?

Let $S \subset R$ be rings, $S$ not necessarily an ideal of $R$, and $S \neq R$. Is there anything that can be said about when $R$ is free as an $S$-module?
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$\mathbb{Z}_m$ is homomorphic image of $\mathbb{Z}_n$

Doesn't this always work as long as $n\geq m$? Can't we get rid of the condition that $n$ is a multiple of $m$? If $n$ is a multiple of $m$, show that $\mathbb{Z}_m$ is homomorphic image of ...
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1answer
94 views

Image of homomorphism from ideal is ideal [duplicate]

Let $A,B$ be rings. If $f:A\rightarrow B$ is a homomorphism from $A$ onto $B$ with kernel $K$, and $J$ is an ideal of $A$ such that $K\subseteq J$, then $f(J)$ is an ideal of $B$: My solution: Let ...
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1answer
48 views

family of ideals

Can somebody explain what exactly is defined to be family of ideals. Is it just an arbitrary collection of ideals of a ring or is there some structure is this family? Thank you
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1answer
193 views

Prove something about Boolean ring

(1) Prove that for a Boolean ring $R$, the following are equivalent: (a)$R$ is artinian; (b) $R$ is noetherian; (c) $R$ is finit; (d) $R$ is semisimple. (2) Prove that if $_RM$ is an artinian or ...
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1answer
48 views

Ideal iff closed with respect to addition and absorbs product

If $A$ is a ring (with unity), prove that $J$ is an ideal of $A$ if and only if $J$ is closed with respect to addition and $J$ absorbs products in $A$. If $J$ is an ideal of $A$, then by ...
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282 views

Find the number of irreducible factors of $x^{63} - 1$

I have to find the number of irreducible factors of $x^{63} - 1$ over $\mathbb{F}_2$ using the $2$-cyclotomic cosets modulo $63$. Is there a way to see how many the cyclotomic cosets are and what is ...
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1answer
114 views

Understanding the homomorphisms from quotient polynomial rings

I'm trying to find all homomorphisms from $\mathbb{R[x]}/(X^2+1)$ to $\mathbb{C}$. I'm using first isomorphisms theorem, as said here Homomorphisms from quotient polynomial rings to some ...
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2answers
169 views

$a+1,a-1$ invertible for nilpotent element

Given a ring (with unity), prove that if $a$ is nilpotent, then $a+1,a-1$ are both invertible. Suppose $a^n=0$. Then $1=1-a^n=(1-a)(1+a+\ldots+a^{n-1})$, so $1-a$ is invertible. If $n$ is odd we can ...
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1answer
124 views

Homomorphisms from quotient polynomial rings to some $\mathbb{Z_n}$

I'm completely lost, my problem is I don't get the gist of a quotient polynomial ring nor ANY homomorphisms between it and some $\mathbb{Z_n}$, much less ALL of them. I know there is something to be ...
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1answer
156 views

Is there an Noetherian ring (commutative) with exactly three prime ideals?

Is there an Noetherian ring (commutative) with exactly three prime ideals $P_i$ which satisfies the following statements? $P_1P_2=0$ and $P_3P_3=0$ $P_1P_3\neq 0$ and $P_2P_3\neq 0$
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2answers
125 views

Proof that the localization $R_S$ is naturally isomorphic to the localization at the saturation $R_{\overline{S}}$?

Localizations have the universal property that if $S$ is a multiplicative subset of a commutative ring $R$, and $i\colon S\to R$ is the canonical embedding, then if $g\colon R\to T$ is any map such ...
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1answer
73 views

Prove that $R$ as a ring is the direct sum of its homogeneous components.

Let $T$ be a simple left $R$-module. Assume that $_RR=\mathrm{Soc}(_RR)$. Prove that the $T$-homogeneous component of $_RR$ is a ring direct summand of $R$, and deduce that as a ring, $R$ is the ...
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2answers
103 views

If $p\in R[X_1,\dots,X_n]$ is irreducible, is it still irreducible in $R[X_1,\dots,X_n,\dots,X_N]$?

It is a known fact that if $R$ is a UFD, then $R[X_1,X_2,\dots]$ is also a UFD, but there is a subtlety that is making me uncomfortable. The standard approach essentially goes something along the ...
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3answers
144 views

How to prove the one-variable calculus definition of derivative extends to $\Bbb C$ *only* because $\Bbb C$ is a field?

I have been told the one-variable calculus definition of derivative extends to $\Bbb C$ only because $\Bbb C$ is a field. See : Higher dimensional analogues of the argument principle? $$ ...