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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Counterexample of “the product of open sets is open in a topological ring”?

Given a topological ring R and U,V open sets, we can show that U+V is an open set due to the fact that $x\mapsto x+y$ is an homeomorphism for every $y \in R$. Since, in general, R is not a division ...
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1answer
13 views

Question about the wording of a Ring Theory problem involving ideals

The homework question is: If $I,J$ are ideals of $R$, let $IJ$ be the set of all sums of elements of the form $ij$, where $i \in I$, $j \in J$. Prove that $IJ$ is an ideal of $R$. The phrase "the ...
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0answers
17 views

How to prove a subset is an ideal

I just started learning about Ring Theory today and I am having some trouble truly understanding and being able to apply certain concepts. The first concept I am having trouble understanding is an ...
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1answer
13 views

Localization of modules and minimal generating sets.

Let $A$ be a ring and $M$ a finite $A$-module; for $p \in \text{Spec} \space A$, write $\mathcal{K}(\mathfrak{p})$ for the residue field of $A_\mathfrak{p}$, and let $\mu (\mathfrak{p}, M)$ denote ...
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2answers
26 views

Rigorous proof that certain ideal is not finitely generated

Let $R = F[x_1,x_2,\ldots]$ (polynomials in an infinite number of indeterminantes) and let $I = \{f \in R : f(0,0,\ldots) = 0)\}$. One can easily see that this is indeed an ideal. The proof for why ...
2
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1answer
30 views

The ideal $\langle x,y \rangle$ in $F[x,y]$ is not principal.

Let $F$ be a field. Apparently we know that $\langle x,y \rangle \neq \langle g(x,y) \rangle$ for any $g \in F[x,y]$. Why is this the case?
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2answers
43 views

Is it possible to turn the set $\mathbb{Hom}(R,S)$ of ring homomorphisms from $R$ to $S$ into a ring?

Is it possible to turn the set $\mathbb{Hom}(R,S)$ of ring homomorphisms from $R$ to $S$ into a ring? Discuss. What I have observed that if I define the multiplication in $\mathbb{Hom}(R,S)$ s.t ...
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1answer
19 views

Krull-Schmidt theorem and internally cancellable modules?

According to this lecture notes (in Lemma2.1) the statement $Ae\simeq Ae^{\prime}\to A(1-e)\simeq A(1-e^{\prime})$ is true for finite dimensional algebras by using Krull-Schmidt theorem. Can anyone ...
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1answer
12 views

Statements about modules (generators and linearly independent sets)

I am trying to prove or disprove with a counterexample the following statements: (i) From every set of generators of a module $M$ one can extract a basis. (ii) Every linearly independent subset of a ...
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0answers
25 views

Is there a valid multiplication for any choice of identity in $C(\mathbb{R})$?

Let $C(\mathbb{R})$ be the ring of continuous functions from $\mathbb{R}$ to $\mathbb{R}$. Its identity with the usual multiplication is $1(x) = 1$. I have two related questions. Firstly, when we ...
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Unity of a subring of $\mathbb Z_{10}$

I've been told that $S=${$[0],[2],[4],[6],[8]$} is a subring of $\mathbb Z_{10}$ with unity $[6]$. How is it true though? $[2][6]=[12]=[2]$, $[4][6]=[24]=[4]$, and so on, isn't it? I realize I'm ...
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0answers
49 views

Looking for a paper by Y. Morita

Does someone have access to the following paper? Y. Morita, Elementary proofs of the commutativity of rings satisfying $x^n=x$, Memoirs Def. Acad. Jap. XVIII (1978), 1-23. MR-Link ...
2
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1answer
44 views

Prove that $S$ is an integral domain and $T$ is not an integral domain.

Let $R = \mathbb{C}[x,y]$ $R^i \subset R$ be the abelian subgroup of $R$ generated by elements of $\mathbb{C}$ times monomials of degree at least $i$ $I = (x^3+x^2-y^2)$ $S = R/I$ $S^i$ be the group ...
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0answers
10 views

Relatively Prime field extensions: Minimal polynomial coefficients

Premise: We are given field extensions $L/K$ and $L(a)/K$ such that degrees $[L:K]$ and $[K(a):K]$ are relatively prime. Question: I would like to show that the minimal polynomial of a over $L$ has ...
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0answers
19 views

Presentation of a module by generators and relations

Let $R:=\mathbb C[T]$. Match the $R$-module with the presentation by generators and relations. $\bullet$$R$-modules: $M:=\mathbb C[T,T^{-1}]$ (Laurent Polynomials)$\qquad$$N:=\mathbb ...
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1answer
61 views

Why only two binary operations?

Ring theory considers things with 2 operations and category theory 101 talks about products and coproducts. I maybe understand why binary operations are more common to look at that trinary, ...
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4answers
37 views

$\mathfrak{a}_{1} + \dots + \mathfrak{a}_{n} = A \Rightarrow \mathfrak{a}_{1}^{r_{1}} + \dots + \mathfrak{a}_{n}^{r_{n}} = A$

I have to prove the following : Let $A$ be a commutative ring with unity and let $\mathfrak{a}_{i}$ be ideals in $A$. Assume that $\mathfrak{a}_{1} + \dots + \mathfrak{a}_{n} = A$. Let $r_{i}$ be ...
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2answers
14 views

non-zero elements in $\mathbb Z_3[i]$ form an abelian group

How shall I show that all non-zero element of $\mathbb Z_3[i]$ form an abelian group of group of order $8$ under multiplication... Please any hint how shall I show this result?
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2answers
33 views

$\mathbb Z[\sqrt d]$ is not a field.

How shall I show that $\mathbb Z[\sqrt d]=\{a+b\sqrt d~~\big|a,b\in \mathbb Z\}$ is not a field? It is an integral domain, so the thing it lacks maybe is every element does not have a ...
2
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1answer
60 views

prove that $ab=1$ implies $ba=1$.

I have a doubt how to prove: If the product of any pair of non-zero elements of $R$ is non-zero , prove that $ab=1$ implies $ba=1$. how shall I make use of the fact : product of any pair of ...
2
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1answer
30 views

Zero divisors in $(\mathbb Z_n,+,*)$

How to understand this : An element $a$ in $(\mathbb Z_n,,+,*)$ is a zero divisor iff $a$ and $n$ aren't coprime... EDIT: Is it also true that an element a in $(\mathbb Z_n,,+,*)$ is a unit iff ...
2
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2answers
35 views

$\mathbb R\oplus \mathbb R$ is an Integral Domain or a Division Ring?

Can anyone help me with these two doubts of mine: Is the ring $\mathbb R\oplus \mathbb R$ an Integral Domain or a Division Ring? My notes state that the ring of Gaussian integers(i.e. $\mathbb ...
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1answer
112 views

Questions related to maximal ideals

In my previous sessional exams, I was asked to prove these two: 1) Find a ring which doesn't have a maximal Ideal. 2) If a ring has unity, then it has a maximal Ideal. About the ...
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2answers
41 views

Proving commutativity [duplicate]

Let $R$ be a ring in which $x^2=x$ for all $x\in R$ where $x^2$ of course denotes $x\cdot x$. a. prove that $x+x=0$, for all $x \in R$ b. prove that $R$ is commutative. I have done part a but how ...
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4answers
59 views

$\mathbb R[x]/\langle x^2+1\rangle$ is a field

How to show that $\mathbb R[x]/\langle x^2+1\rangle$ is a field. I wrote the representation of $\mathbb R[x]/\langle x^2+1\rangle$ =$\{a+bx+\langle x^2+1\rangle\big|a,b\in \mathbb R\}$. Now how ...
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1answer
47 views

Show structure of a commutative ring in a tensor product

I need some help with this: Let $R$ be a commutative ring and $S$ and $T$ be commutative $R$-algebras. Show that $$ S \otimes T $$ has the structure of a commutative ring with multiplication: $$ (s ...
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1answer
33 views

Doubt regarding zero elements in factor ring :$\mathbb Z[i]/\langle3-i\rangle$

I have the factor ring $\mathbb Z[i]/\langle3-i\rangle$ and am asked to find elements zero in this ,they are $0,3-i,i(3-i),(3-i)+i(3-i)$. But I can't understand how do we guarantee these are the only ...
2
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3answers
59 views

Proving ring $R$ with unity is commutative if $(xy)^2 = x^2y^2$

I tried several methods to solve this but couldn't get through. Now the solution in almost all the textbooks goes like this. First take $x$ and $y+1$ so that $ (x(y+1))^2 = x^2(y+1)^2 => xyx = ...
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1answer
22 views

An Ideal which is Maximal additive subgroup is a Maximal Ideal

How should I prove this: Any Ideal which is a Maximal additive subgroup is also a Maximal Ideal . any idea how to prove it..
1
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1answer
25 views

$R/J(R)$ not semisimple Artinian

I search for a ring $R$ with Jacobson radical $J(R)$ such that $R/J(R)$ is not semisimple Artinian. Being a finitely generated module over itself, $R$ would have infinite hollow dimension due to ...
2
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2answers
27 views

Do integers modulo $n$ minus $\frac n 2$ (i.e. signed integers) still form a commutative ring?

This is related to this (closed) question on programmers.sx. I'm looking into the properties of (64bit signed) computer integers. My question is whether they do form a commutative ring as ...
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1answer
32 views

A question regarding the number of generators of an ideal [duplicate]

Let $I$ be an ideal in $\mathbb{C}[x_1 ,x_2 ,x_3 ,x_4 ]$ such that $I$ is generated by $x_1 x_3$, $x_2 x_3$, $x_1 x_4$, and $x_2 x_4$. How to show that this I cannot be generated by two ...
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1answer
23 views

Have to show that $q(x) \in$ $Z[x]$

Let $f(x) \in$ $Z[x]$ with $c(f)$=$1$ and $f$ is non constant. Now suppose $h(x) \in$ $Z[x]$ be such that $h(x)$=$f(x)q(x)$ where $q(x) \in$ $Q[x]$ . Then I have to show that $q(x) \in$ $Z[x]$ ...
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2answers
47 views

Show that the ring of continuous functions $f:\mathbb R\to\mathbb R$ is not Noetherian

Prove that the ring of continuous functions $f:\mathbb R\to\mathbb R$ is not Noetherian. I know that to be Noetherian, every ideal is generated by finitely many elements or equivalently R ...
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0answers
11 views

Ideal from ring of fraction

Given $R$ is a commutative ring with $1$ and $D$ is multiplicatively closed containing $1$, I want to show that any ideal of $D^{-1}R$ is of the form $D^{-1}I$, where $I$ is an ideal in $R$. I have ...
2
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0answers
45 views

UFD and relatively prime elements

I've found the following statement at page 9 of Griffiths, Harris "Principles of Algebraic Geometry": Proposition. If $R$ is a UFD and $u$, $v \in R[t]$ are relatively prime, then there exist ...
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1answer
22 views

If I define $ I.J=\{ij : i \in I $ & $ j \in J \} $. Then prove that it is not necessrily an ideal, where $I,J$ are ideals in a ring $R$. [duplicate]

If I define $ I.J=\{ij : i \in I $ & $ j \in J \} $. Then prove that it is not necessrily an ideal, where $I,J$ are ideals in a ring $R$. I have found one counter example in $R[x,y,z]$ for ...
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0answers
16 views

Direct product of nontrivial rings can never be an integral domain or a field [duplicate]

I saw this claim in Pinter's A book of Abstract Algebra. I cannot understand. If the two rings are both integral domains, the direct product should be an integral domain. If the two rings are both ...
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1answer
25 views

If $\cap_{j=1}^{n}I_{j} \subseteq P$ for any ideals $I_1,I_2,..I_n$, then $I_j \subseteq P$ for some $j$

$P$ is a prime ideal if $P$ satisfies the following : If $\bigcap\limits_{j=1}^{n}I_{j} \subseteq P$ for any ideals $I_1,I_2,..I_n$, then $I_j \subseteq P$ for some $j$, where $R$ is a commutative ...
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0answers
20 views

prime ring and the proof of it's property [on hold]

for a ring R , show that the folloeing conditions are equivalent: (1)R is a prime ring. (2)for elements a, b belongs to R ,we have aRb=(0)implies that a=0 or b=0.
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2answers
48 views

Sum and product of comaximal ideals

Let $R$ be a commutative ring with unity. If $R=I_{i}+I_{j}$, for all $i\ne j$, where $I_1,I_2,...,I_n$ are ideals of $R$, I want to show that $$R=I_{n}+I_{1}I_{2}\cdots I_{n-1}.$$ I started off ...
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2answers
20 views

Questions related to the concept of $k$-algebras

I am reading about modules and some days ago I've worked on some exercises related to $k$-algebras. The definition I've seen of $k$-algebra is that it is a field $k$ and a ring $A$ together with a ...
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0answers
31 views

Euclidean Domains in Ring Theory [closed]

Prove that $\displaystyle \mathbb{Z}\Bigg[\frac{1+\sqrt{-3}}{2}\Bigg]$ is a Euclidean Domain.
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0answers
12 views

Is a field a PID? [duplicate]

How can I prove that a field is a PID? I can prove that a field is an Integral Domain, but stuck in proving that every ideal is principal.
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1answer
24 views

Congruences in Algebra [on hold]

I have a question regarding a particular statement of a given Ring Theory problem. It is "$x$ is unique mod $n=n_1n_2...n_k$". Can anyone please tell me the meaning of this statement?
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0answers
43 views

Quotient Field of an Integral Domain

The question is: Let $n$ be a positive integer and let $D$ be the subset of all rational numbers of the form $$\frac{a}{n^k}$$ with $a \in \mathbb{Z}$ and $k$ any positive integer. Show that $D$ ...
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1answer
51 views

Some questions about finite rings [closed]

Let $K$ be finite associative ring with nonzero multiplication. Are the following statements true: If for an element $a \in K$ there exist an element $b \in K$ such that $ax=b$ for all nonzero ...
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0answers
28 views

Surjective ring homomorphism from $M_n(R)$ to $M_n(R/I)$ where $R$ is a ring and $I$ is an ideal for R?

I'm looking for such a surjective homomorphism. I was thinking of starting from the canonical surjection from $R$ to $R/I$ and induce one but somehow I get stuck... Can you help me? Thank you very ...
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2answers
41 views

The krull dimension of $\Bbb{Z}$ and artinian rings

On page thirty of Matsumura, it says that $\Bbb{Z}$ has krull dimension 1 because every prime ideal is maximal. I understand this because for any prime p you have $0 \subset p$. However, for artinian ...
5
votes
3answers
45 views

Show some polynomial is irreducible over the field of 7 elements.

I have to show that the polynomial $x^4+x^3+x^2+x+1$ is irreducible over the field $F_7$. It doesn't have roots in $F_7$, but I can't show it does not have degree two irreducible factors in $F_7[x]$. ...