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
105 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. ...
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0answers
184 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
85 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
57 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 ...
4
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1answer
87 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 ...
1
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1answer
128 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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2answers
254 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
118 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 ...
1
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1answer
70 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
96 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 ...
2
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1answer
185 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
54 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 ...
4
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2answers
59 views

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 ...
0
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1answer
61 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 ...
4
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1answer
88 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
124 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 ...
2
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2answers
91 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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3answers
66 views

$\mathbb{Z}_m$ 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 ...
0
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1answer
80 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
34 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
136 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 ...
0
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1answer
39 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 ...
4
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2answers
241 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
100 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
125 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 ...
3
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1answer
94 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 ...
2
votes
1answer
150 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
74 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 ...
2
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1answer
63 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
86 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
130 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? $$ ...
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2answers
158 views

Is $2\mathbb{Z}/4\mathbb{Z}$ NOT a field?!

According to wikipedia: If $R$ is a unital commutative ring with an ideal $m$, then $k = R/m$ is a field if and only if $m$ is a maximal ideal. In that case, $R/m$ is known as the residue field. ...
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7answers
378 views

What's the rationale for requiring that a field be a $\boldsymbol{non}$-$\boldsymbol{trivial}$ ring?

The title pretty much says it all. Of course, one answer (IMO unsatisfactory) to such questions goes something like "a definition is a definition, period." In my experience, mathematical definitions ...
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1answer
140 views

When is a vector space (over field $K$) also a ring (with subring $K$)?

(Apologies in advance for the very naive question. I'm just learning about all this. Also, for the sake of expedience, below I use the word "ring" when it would more correct for me to use ...
3
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0answers
69 views

Graded Betti Numbers of a Graded Ideal with Linear Quotients

Exercise 8.8(a) in Monomial Ideals by Herzog and Hibi: Let $I\subset S=K[x_{1},...,x_{n}]$ be a graded ideal which has linear quotients with respect to a homogeneous system of generators ...
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0answers
75 views

Graded Betti Numbers of a Stable Monomial Ideal

Exercise 8.8 in Monomial Ideals by Herzog and Hibi: Let $I\subset S=K[x_{1},...,x_{n}]$ be a stable monomial ideal with $G(I)=\{u_{1},...,u_{m}\}$ and such that for $i<j$, either ...
0
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2answers
108 views

How to show that $x$ becomes a root of $p(x)$ in $F[x]/(p(x))$

$F$ is a field, $p(x)$ is irreducible polynomial at $F[X]$. $K=F[X]/\left<p(x)\right>$. For every $a\in F$ we will mark: $\bar{a}=\left<p(x)\right>+a$. Now, the question is: How do I show ...
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2answers
176 views

The proof that a finite field has a prime power order

I don't seem to grasp the proof. First we construct a vector space over a subfield with prime order $p$ where $p$ is the characteristic of the field . As the field is finite , the vector space will be ...
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2answers
101 views

How to find a polynomial product that give me $x^6+1$

I need to find a polynomials product that give me $x^6+1$ at $\mathbb{R}[X]$ and at $\mathbb{C}[X]$. I need that the product will be of irreducible polynumials... Thank you!
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4answers
68 views

Consider $S = \{(n,n)\mid n\in \mathbb{Q}\}$. Prove $S$ is a subring of $\Bbb Q \times \Bbb Q$ but not an ideal

Consider $S = \{(n,n)\mid n\in \mathbb{Q}\}$. Prove $S$ is a subring of $R$ but not an ideal in $R$, where $R = \mathbb{Q} \times \mathbb{Q}$. I don't know how to do this one... any help would be ...
6
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1answer
175 views

Minimal counterexamples of the isomorphism problem for integral group rings

The isomorphism problem for integral group rings asks if two finite groups $G,H$ are isomorphic when their integral group rings $\mathbb{Z}[G]$, $\mathbb{Z}[H]$ are isomorphic. Quite a lot has been ...
0
votes
1answer
80 views

coprime elements

Let $R$ be a ring, then two elements $I,J$ are coprime, if $RJ+RI=R$ or in other words, if there exist $r_1,r_2 \in R$ such that $r_1I+r_2J=u$, where $u$ is a unitity in $R$. Now let $\mathbb{Q}$ be ...
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3answers
140 views

Can Boolean ring without unit be embedded into a boolean ring?

While going through a book (Lectures on Boolean algebra, Halmos) I got struck at the following question : Prove that every Boolean ring without a unit can be embedded in a Boolean ring with a unit. ...
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2answers
71 views

Non-unital commutative ring without non-prime ideals?

Does there exist a non-unital commutative ring such that all its proper ideals are prime? Note also that that equipping the abelian group $\mathbb Z/p\mathbb Z$ with trivial multiplication $xy=0$ for ...
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2answers
56 views

Ideals of a subring of $M_{2\times 2}(\mathbb{R})$

Define $$A:=\left[\begin{array}{cc}\mathbb{R}&\mathbb{R}\\ 0&\mathbb{R}\end{array}\right]=\left\{\left[\begin{array}{cc}a&b\\ 0&c\end{array}\right]:a,b,c\in\mathbb{R}\right\}$$ Prove ...
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1answer
119 views

Proving that set is (or is not) a field

Let $P = \{a + b\sqrt[3]3 + c\sqrt[3]9, a, b, c \in \Bbb Z \}$ It is easy to prove that $(P, +, \cdot)$ is a ring considering ordinary addition and multiplication. How to prove that this set is or is ...
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0answers
53 views

Hochschild homology with trivial coefficients: how to make $K$ an $M_n(K)$-module

Let $R$ be a ring, $A$ an associative $R$-algebra, and $M$ an $A$-$A$-bimodule. Then the Hochschild homology of $A$ with coefficients in $M$, denoted $HH_\ast(A)$, is the homology of the chain complex ...
2
votes
2answers
381 views

homomorphism image of a maximal ideal of a ring

Let $R$ and $S$ be commutative rings with $1$ and $\phi: R\rightarrow S$ be a surjective ring homomorphism. Then for an arbitrary maximal ideal $I$ of $R$, does $\phi(I)$ have to be maximal in $S$? ...
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votes
4answers
108 views

Let $K$ be a field and $f(x)\in K[x]$. Prove that $K[x]/(f(x))$ is a field if and only if $f(x)$ is irreducible in $K[x]$.

Let $K$ be a field and $f(x)\in K[x]$. Prove that $K[x]/(f(x))$ is a field if and only if $f(x)$ is irreducible in $K[x]$. How to prove? I really have no idea... Thank you a lot.