Articles of abstract algebra

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 […]

Lines in the projective plane

In my lecture notes we have the following: The set $$\mathbb{P}^2(K)=\{[x, y, z] | (x, y, z) \in (K^3)^{\star}\}$$ is called projective plane over $K$. There are the following cases: $z \neq 0$ $$\left [x, y, z\right ]=\left [\frac{x}{z}, \frac{y}{z}, 1\right ]$$ $$\mathbb{P}^2(K) \ni \left [x, y, z\right ] \to \left (\frac{x}{z}, \frac{y}{z}\right ) \in […]

How many elements there exist in polynomial quotient ring $\mathbb{Z}_5/(X^2+1)?$

How many elements there exist in quotient ring $\mathbb{Z}_5[X]/(X^2+1)$? I’m learning polynomial ring. However I cant’ completely understand the number of elements. I think that the elements are contained as follows. \begin{align} \mathbb{Z}_5[X]/(X^2+1)=\{&0,1,2,3,4,x,x+1,x+2,x+3,x+4,\\ & 2x,2x+1,2x+2,2x+3,2x+4,3x,3x+1,3x+2,3x+3,3x+4,\\ & 4x,4x+1,4x+2,4x+3,4x+4,x^2 \} \end{align} There are 26 elements in total, is it correct?

How to show that $(Y- X^2, Z – X^3) \subseteq k$ is a prime ideal?

I suppose that $k$ is an algebraically closed field (actually, my goal is to show $\mathcal{I}(\mathcal{V}(Y- X^2, Z – X^3)) = (Y- X^2, Z – X^3)$). (But I think algebraically closed is not necessary to show $(Y- X^2, Z – X^3) \subseteq k[X,Y,Z]$ is a radical ideal…) My strategy is to show $k[X,Y,Z]/(Y- X^2, Z […]

The value of the rational “Möbius”-like transformations at infinity

If $\mathbb F$ is some field, the group $PSL(2, \mathbb F)$ consists of the mappings $$ x \mapsto \frac{ax + b}{cx + d} $$ with $a,b,c,d \in \mathbb F$ and $ad – bc = 1$. These mappings are defined over the extended field $\mathbb F_{\infty} := \mathbb F \cup \{\infty\}$ (or the so called projective […]

Smallest Graph that is Regular but not Vertex-Transitive?

I’m trying to find the smallest graph that is regular but not vertex-transitive, where by smallest I mean “least number of vertices”, and if two graphs have the same number of vertices, then the smaller is the one with the lower number of edges. I currently have that the smallest such graph is the disjoint […]

In a finitely generated $k$-algebra, the nilradical is $0$ iff the Jacobson radical is $0$.

I was solving an exercise in Vakil’s notes Foundations of Algebraic Geometry 3.6.K, and eventually proved the following statement: Let $\mathscr{A}$ be a finitely generated $k$-algebra, where $k$ is any field. Then $\mathscr{N}(\mathscr{A}) = 0\iff \mathscr{R}(\mathscr{A}) = 0$. (All $k$-algebras are commutative here.) Here is my proof for the hard direction: Let $0\ne f\in \mathscr{A}$, […]