Articles of abstract algebra

Why is $\mathbb{Z}/n\mathbb{Z}$ quasi-Frobenius?

When surfing the wiki, I found the definition of Quasi-Frobenius rings $R$ is quasi-Frobenius if and only it satisfies the following equivalent conditions: All right (or all left) R modules which are projective are also injective. All right (or all left) R modules which are injective are also projective. Then, it mentions that the quotient […]

Isomorphism of Hom sets

If $R$ is a ring and $I\subseteq R$ is an ideal, I want to see why $$\textrm{Hom}_R(I, R/I) \cong \textrm{Hom}_R(I/I^2, R/I)$$ I read somewhere that this is true because of the action of any element in $I$. I’m having difficulty seeing what this means explicitly. Can someone spell it out?

If $K/\mathbb{Q}$ is finite, then $K$ contains finitely many $n$th roots of unity

This is exercise 13.6.5 from Dummit and Foote. Let me first note that I realize there is already a post on this question here. However, I am a bit confused about the explanation given in that post, and I want to discuss my confusion about it here. The following is the beginning of an argument […]

Exact sequence and torsion

I’ve come across another exact sequence, where (I guess) I need to deduce the result using some properties of torsion. I am calculating the homology of the Klein bottle using attaching maps. I start by defining $\Phi:I \times I \to K$ as the natural map and denote $\partial(I \times I)$ as the boundary, then let […]

Basic constructions for graded algebras.

I’m reading about the Weil algebra of a Lie group and it involves some constructions I’m not very familiar with, for instance the “free graded-commutative graded algebra on $a_1…a_n$ with degrees $deg(a_i)$.” Does anyone know a good source for basic graded-algebra constructions like this? Anything i find is either too basic or too advanced to […]

How does a surjection from $\pi_1(\Sigma_g)$ to a group $Q$ determine an element of $H_2(Q,\mathbb{Z})$?

Let $\Sigma_g$ be a closed orientable surface of genus $g$ on page 36 of this paper. It is asserted that for a finite group $Q$, a homomorphism $\pi_1(\Sigma_g)\rightarrow Q$ determines an element in $H_2(Q,\mathbb{Z})$. How does this work? This is especially confusing to me, since they don’t even specify the action of $Q$ on $\mathbb{Z}$, […]

Question concerning a faithful module over an Artinian ring

Let $A$ be an Artinian ring and $M$ an $A$-module. Is it true that $M$ is faithful if and only if there is an exact sequence of the form $0\rightarrow A \rightarrow M^r$ for some natural $r$ ? Thanks for the help.

Polynomial Interpolation and Data Integrity

This question is about polynomial interpolation and security. Please consider a scenario where we have a polynomial $f$, one of whose roots is $a$. We evaluate it at some $\textbf{x}=(x_0,\ldots,x_n)$ and this yields $\textbf{y}=(y_0,\ldots,y_n)$, where $y_i=f(x_i)$. We shuffle the elements in $\textbf{y}$. Then we give $\textbf{x}$ and the shuffled $\textbf{y}$ to an untrusted server. The […]

If $s$ is a multiple of both $a$ and $b$, then $s$ is a multiple of $\operatorname{lcm}(a,b)$

I came across this as I was doing work for one of my classes. We just use this property, proved presumably in number theory, which we didn’t need to take. Could someone help me? Prove that if $a$, $b$ are positive integers, and $m=\operatorname{lcm}(a,b)$, and if $s$ is a multiple of both $a$ and $b$, […]

Surjective Maps and right cancellation

I’m working through Jacobson’s Basic Algebra I, and I have a question about Exercise 3 in Section 0.2. The first part asks the reader to “Show that $ S \xrightarrow{\rm \alpha}T $ is surjective if and only if there exist no maps $ \beta_1, \beta_2 $ of $T$ into a set $U$ such that $\beta_1 […]