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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Equivalent definitions of an algebra over a ring

I have seen two different definitions from several sources of an algebra over a ring. From Wikipedia: Let R be a commutative ring. An R-algebra is an R-Module A together with a binary operation ...
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1answer
33 views

What does $[5^{2000}]$ equal to in $\mathbb{Z}_4$?

Question: What does $[5^{2000}]$ equal to in $\mathbb{Z}_4$? My proof: (which I doubt whether its correct or not since it doesn't use the hint in the book) $[5^{2000}]=([5])^{2000}$ Since $5 \equiv ...
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Prove polynomial f(x) irreducible over the integers.

Consider the polynomial $f(x) = (x − 1)(x − 2)···(x − (n - 1))(x − n) + 1$, for some $n \in \mathbb{N}$. Prove that $f(x)$ cannot be reduced to the form $f(x) = a(x) \cdot b(x)$ for polynomials $a, b$ ...
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14 views

Basis of the ring $B=End_R(R^{(\mathbb N)})$

Let $B=End_R(R^{(\mathbb N)})$. Define $u,v \in B$ as $$u(e_{2_{i+1}})=0, u(e_{2_i})=e_i$$$$v(e_{2_{i+1}})=e_i,v(e_{2_i})=0$$ Prove that $\{u,v\}$ is a basis of $B$ as a $B$-module. I've already ...
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1answer
20 views

ring morphism from a group ring to another ring

I've read that if $S$ is a commutative ring, then $Hom_R(R[G],S)=Hom_R(R,S)\times Hom_{Gr}(G,\mathcal U(S))$. I've tried to show this equality but I couldn't. If $\phi: R[G] \to S$ is a ring ...
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1answer
19 views

Graded ring localization. Why is this function bijective?

From Hartshorne, chapter II.2, proposition 2.5-b. If R is a graded ring and a is a homogenous ideal, then the function defined as $\phi(a) = (a.R_f)\cap R_{(f)}$ is a bijection. Where $R_f$ is ...
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2answers
19 views

How to show that ideal is prime in $\mathbb{R}[x,y,z]$ modulo some other ideal

Let $R:=\mathbb{R}[x,y,z]$ and $g:=x^2+y^2-z^2\in R$. I would like to know how to show that $(x,y-z)/(g)$ is a prime ideal in $R/(g)$, and whether it is maximal or not. Thanks for the help!
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1answer
18 views

Polynomial rings of two variables

Prove that $(x,y)$ is not a principal ideal in $\mathbb{Q}[x,y]$. Here what is the definition of $(x,y)$? I don't know how to start the solution since I don't know the meaning of $(x,y)$.
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Noether normalization for $k[x]_{x}$

According to the Noether normalization theorem, there exists a $k[t]$ where $t$ is an indeterminate and $k[t]\subseteq k[x]_{x}$ is a $k$-algebra extension so that $k[x]_{x}$ is a finitely generated ...
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Is this ring a well known ring and if so how is it called?

I just had this thought when I was thinking how I was introduced to the concept of number in primary school and I came upon the conclusion that the numbers we were taught to manipulate (adding, ...
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1answer
21 views

Question on Wedderburn components of $\mathbb C[G]$

$G$ is a finite group. Wedderburn's theorem says that $\mathbb{C}G\cong R_1 \times\cdot \cdot \cdot R_r$ as rings where $R_i=M_{n_i}(D^i)$ is the ring of $n_i\times n_i$ matrices over a division ring ...
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2answers
15 views

Quick way to determine existence of integral root of a polynomial in one variable

Suppose $p(x) \in \mathbb{Z}[x]$ and if there exist $b \in \mathbb{Z}$ s.t. $p(b)=0$, then $x-b|p(x)$. The other technique can be to put all $b \in \mathbb{Z}$. But this require to check every $b \in ...
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0answers
15 views

direct sum of modules and generator subset

I am trying to solve the following problem: Let $(M_i)_{i \in I}$ be an infinite family of non zero modules and $S$ a system of generators of $\bigoplus_{i \in I}M_i$. Prove that the cardinal of $S$ ...
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43 views

Division Rings and trivial ideals

I'm stuck in the following exercise, I guess this is easy but I would appreciate someone's tip.... I have to prove the following: If $R$ is a right simple ring (the unique right ideals are $R$ and ...
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1answer
14 views

Expression for polynomial in $k[x,y]$.

Let $k$ be any field. For any polynomial $f \in k[x,y]$ apparently one can write $f(x,y) = f_0(x) + f_1(x)y + (x^3 - y^2)g(x,y)$. Why is this the case?
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1answer
20 views

Characterize semisimple rings with a unique maximal ideal

Problem Characterize the semisimple rings $R$ that contain a unique maximal ideal. I am not so sure what to do here. I know that a ring $R$ is semisimple if and only if all $R$-modules are ...
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1answer
26 views

Prove or disprove statements about modules

I am trying to determine if the following statements are true or false (i) There are free modules with non zero elements $x$ such that $\{x\}$ is linearly dependent. (ii) There are non free modules ...
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1answer
40 views

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

Given a topological ring $R$ and $U,V$ open subsets, we can show that $U+V$ is an open subset due to the fact that $x\mapsto x+y$ is a homeomorphism for every $y \in R$. Since, in general, $R$ is not ...
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1answer
18 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
24 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
18 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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3answers
44 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 ...
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1answer
33 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
50 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
26 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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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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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 ...
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1answer
55 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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20 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
70 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
39 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
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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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36 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 ...
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1answer
62 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 ...
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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 ...
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2answers
38 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
115 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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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
62 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
49 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
34 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 ...
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3answers
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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..
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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 ...
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2answers
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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
34 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
24 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
50 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 ...