Tagged Questions

This tag is for questions about rings, which are a type of algebraic structure studied in abstract algebra and algebraic number theory.

59 views

58 views

Compute the distance between two elements in a ring

Given a ring of size $n = 2^m$, starting with element $0$ to element $n-1$, what general formula gives the distance between two arbitrary elements $i$ and $j$? Note that the distance between the ...
75 views

Find a non-principal ideal in $\Bbb Z [2i]$.

Find a non-principal ideal in $\Bbb Z [2i]$. I think it might be $(1+2i,1-2i)$, but have problems with proving this. I know that $|1+2i|=|1-2i|=5$. Moreover, there are only 6 elements with non-...
293 views

Find prime ideals of the ring $\Bbb Z [ \sqrt[3]2]$ which contain $5$

Find all prime ideals $p$ of the ring $R= \Bbb Z [ \sqrt[3]2]$ such that $5 \in p$ and find $R/p$ for all of them. I know that of course $R$ is prime and $R/R = \{0 \}$. Unfortunately I have no ...
61 views

Ring isomorphism

What is the simplest form of $\mathbb{Z}[X]/(X^2+2X)$ ? I tried to use the first isomorphism theorem, but I have problems if finding a proper map $\phi$ such that $\ker\phi=(X^2+2X)$.
147 views

Cardinality of the ring $F_3[x] / (x^2-x+1)$

How would I find the number of elements of the ring $F_3[x] / (x^2-x+1)$? I know that $x^2-x+1$ is not prime/irreducible, since gcd($x^2-x+1$, $x^3-x^2-1$) = 3. Can anyone provide some tips?
114 views

Example of a compact module which is not finitely generated

Let $R$ be a ring and $M$ be an $R$-module. Definition: $M$ is called compact if $\text{Hom}_R(M,-)$ commutes with direct sums, that is, if for any set $I$ and any $I$-indexed family of $R$-...
109 views

LCM of Polynomials

I know for integers we have $$\operatorname{lcm}(n,m) = \frac{nm}{\gcd(n,m)}$$ Does this hold for polynomials? i.e. $\operatorname{lcm}(f(x),g(x)) = \dfrac{f(x)g(x)}{\gcd(f(x),g(x))}$
75 views

45 views

$F$ is isomorphic to $\Bbb Z_p$ for some prime number $p$. [duplicate]

Suppose $F$ is a field and there is a ring homomorphism from $\Bbb Z$ onto $F$. Then show that $F$ is isomorphic to $\Bbb Z_p$ for some prime number $p$. I am facing difficulty to do the proof. I ...
35 views

Ideals and quotient rings [duplicate]

I am trying to show that (R/I)/(J/I) is isomorphic to R/J I and J are both ideals of the ring R, and I is a subset of J. How do I begin this proof?
60 views

Are these topologies equivalent?

Consider the space $(\mathbb C^2 , τ_1 )$ where $τ_1$ is the product topology on $\mathbb C^2$ with $\mathbb C$ having the Zariski topology i.e. closed sets indexed by $p(x) \in \mathbb C[x]$ are ...
43 views

32 views

Example of a Dedekind Finite Ring Which is Not Stably Finite

I know that there is a Dedekind Finite Ring which is not Stably Finite. Shephardson has given such an example. I need some different example. Can anyone supply me another example?
25 views

337 views

Elements of an annihilator induced by a matrix

An annihilator is defined as $Ann_R(M) = \{r \in R | rm=0 \forall m \in M \}$. However, I read that the minimal polynomial of a matrix $A$ generates $Ann_A(V)$. But, I do not understand; what could be ...
64 views

Image of ring homomorphism $\phi : \mathbb{Z}[t] \to \mathbb{Q}$?
Here is a problem I face practicing the theory of rings: Define $\phi : \mathbb{Z}[t] \to \mathbb{Q}$, a ring homomorphism (it does map $1$ to $1$). I'm trying to show that if $\phi(t)=\frac{u}{v}$...
Consider the space $(\mathbb{C^2},T)$ where $T$ is the product topology on $\mathbb{C}$ with $\mathbb{C}$ having Zarisky topology. Now let $T_2$ defines another topology on $\mathbb{C}^2$ with open ...