Articles of abstract algebra

Basic question regarding a finitely generated graded $A$-algebra

Let $S = \oplus_{n \geq 0} S_n$ be a graded ring. Let $S$ be a finitely generated $A$-algebra, where $A = S_0$, a commutative ring with unity. Then there exists $t_1, .., t_M$ homogeneous elements of positive degree that generate $S$ over $A$. It then follows that $S$ is isomorphic to $k[x_1, …, x_M]/I$ as […]

if the kernel of an endomorphism is a direct summand, doesn't the image have to be too?

Let $M$ be a module and $\phi: M\longrightarrow M$ be an endomorphism. Let $\ker \phi$ be a direct summand of $M.$ Does it imply that $\mbox{im}\,\phi$ is a direct summand of $M$ too? I have not seen an explicit statement in any text have read but it seems quite clear from what I have seen […]

Necessary and Sufficient Condition for $\phi(i) = g^i$ as a homomorphism – Fraleigh p. 135 13.55

Let $g \in \text{ group } G $ and $n \in N$. Let $\phi : \mathbb{Z_n} \rightarrow G$ be defined by $\phi(i) = g^i$ for $0 \le i \le n$. Give a necessary and sufficient condition (in terms of g and n) for $\phi$ to be a homomorphism. Prove your assertion. My $g =$ solution’s […]

zero divisors of ${\bf Z}_n$

Consider the following proposition: A nonzero element $m\in{\bf Z}_n$ is a zero divisor if and only if $m$ and $n$ are not relatively prime. I don’t know if this is a classical textbook result. (I didn’t find it in Gallian’s book). For the “only if” part, one may like to use the Euclid’s lemma. But […]

$S_n$ acting transitively on $\{1, 2, \dots, n\}$

I am reading Dummit and Foote, and in Section 4.1: Group Actions and Permutation Representations they give the following example of a group action: The symmetric group $G = S_n$ acts transitively in its usual action as permutations on $A = \{1, 2, \dots, n\}$. Note that the stabilizer in $G$ of any point $i$ […]

Existence of the least common multiple in a Unique Factorization Domain

Let $x,y \in R$, where $R$ is a unique factorization domain. Let $P$ be the set of all representatives in each class of associate irreducible elements of $R$. Then, suppose $x,y\in R$ are nonzero, nonunit elements (First Question: How would this proof change if either $x$ or $y$ is zero or is a unit of […]

Norm of powers of a maximal ideal

Let $A$ be a integral domain and $M$ a maximal ideal in $A$ such that the quotient $A/M$ is a finite ring (and thus a finite field). Is it true, in general, that $$|A/M^k|=|A/M|^k \quad (k\in\textbf{N})\ ?$$ Edit. (Counter-example in the answers, thanks to Jendrik Stelzner and Bib-lost). Nevertheless, I have the feeling that this […]

Integral Domains

If $R$ is an integral domain and a subring $S$ has identity $1_S$, how would you show that $1_S=1_R$ (here $1_R$ is the identity of the ring $R$)? I am unsure about what an integral domain really is and how the subring comes into play here.

“Compositional roots” of functions, how to define them and how many are there?

Assume I am interested in solving $$(\underset{k \text{ times}}{\underbrace{g\circ \cdots \circ g)}}(x) = g^{\circ k}(x) = f(x)$$ That is, $g$ is in some sense a function which is a $k$:th root to applying the function $f$. Applying $g$ $k$ times starting with the number $x$ does the same thing as applying $f$ once. I suspect […]

Induced map on simplices from order-preserving maps between finite ordinal numbers

Let $\Delta$ be the category of finite ordinal numbers with order-preserving maps, i.e., $\Delta$ consists of objects strings $[n]: 0→1→2→⋯→n$. A morphism $f:[n]→[m]$ is an order-preserving function (a functor) and we can think of the morphism like diagrams where arrows don’t cross. For an arrow $[n] \overset{\Theta}{\to} [m]$ we get an induced map $| \Delta^n […]