Articles of abstract algebra

Is every equivalence of monoidal categories a monoidal equivalence?

Let $C$ and $D$ be monoidal categories. Let $T:C \to D$ be a functor that gives the equivalence of $C$ and $D$ as just categories. My question is whether such an equivalence $T$ is always a monoidal equivalence or not. If this is true, could you give me a proof? Thank you.

Irreducibility in $\mathbb{F_2}$ and field extensions

Here’s an exercise I found: 1) Show that $$f(X):=X^3+X+1$$ $$g(X):=X^3+X^2+1 $$ are irreducible in $\mathbb{F_2}[X]$. 2) Let $a, b \in \overline{\mathbb{F_2}}$ with $f(a)=g(b)=0$. $$\mathbb{F_2}\subseteq\mathbb{F_2}(a),\mathbb{F_2}(b)$$ are field extensions. Show that $\mathbb{F_2}(a)$ and $\mathbb{F_2}(b)$ are isomorphic $\mathbb{F_2}$-algebras. 1) is pretty easy to show: Since $\mathbb{F_2}$ is a field, a polynomial of degree $3$ can only be divided […]

Definibility of $\mathbb{Z}$ in product rings

If $R$ is a product ring whose factors are in a finite number and are all quotients of $\mathbb{Z}$ (that is, either $\mathbb{Z}$ or $\mathbb{Z}_n$’s ), is it a sufficient and necessary condition for the diagonal image of $\mathbb{Z}$ to be undefinable in it by a finite first-order formula, that there be at least two […]

Show that $M = \{a + b\sqrt{2}\mid 5\mid a, 5 \mid b\}$ is a maximal ideal in $R = \mathbb{Z}$ and find $R/M$.

I came across this question as part of my self-study of abstract algebra and so I would prefer answers suitable for a beginner. First I established that the set $$M = \{a + b\sqrt{2}\mid a, b\in \mathbb{Z}, 5\mid a, 5 \mid b\}$$ is an ideal in the ring $R = \mathbb{Z}[\sqrt{2}]$ and this is more […]

Tensor products of maps

Let $V, W, U, X$ be $R$-modules where R is a ring. At what level of generality, if any is it true that the maps (I always mean linear) from $V \otimes W$ to $U \otimes X$ can be identified with $L(V, U)\otimes L(W, X)$ where $L(., .)$ is the space of maps, via the […]

Let $G=GL(2,\mathbb{Z}/5\mathbb{Z})$, the general linear group of $2 \times 2 $ matrices with entries from $\mathbb{Z}/5\mathbb{Z}$

Let $G=GL(2,\mathbb{Z}/5\mathbb{Z})$, the general linear group of $2 \times 2 $ matrices with entries from $\mathbb{Z}/5\mathbb{Z}$. i) List the elements of the centre of $G$. ii) Compute the order of $G$. iii) Give an example of an element of $G$ of order $4$, and of an element of $G$ of order $8$. iv) Let $x$ […]

Sum of free submodules of a module over a PID

It’s possible to produce an example of an integral domain $R$ and a free $R$-module $M$ with free submodules $L, L’$ such that $L+L’$ is not free. We can take $R=M=K[x,y]$ , $L=\left<x\right>$ , $L’=\left<y\right>$. If $R$ is a PID and $M$ is free $R$-module, then for every pair of submodules $L, L’$ of $M$ […]

Condition to be a group.

Let $G$ be a semigroup. I’m showing that $G-group \iff [ \ \exists_{e\in G} \forall_{a\in G}: ea=a\ ] $ and $ [\ \forall_{a\in G}\exists_{a^{-1}\in G}: a^{-1}a=e \ ] $ “$\Rightarrow$” is obvious. “$\Leftarrow$” This is how I do it: Let $a \in G$. $ea=a$ $aea=aa$ $(ae)a=aa$ Is it true that this implies $ae=a$? How’s that […]

Associativity of norms in inseparable extensions

Let $K$ be a field. Let $L/K$ and $E/L$ be finite extensions. Let $α$ be an element of E. Let $N_{E/K}(α)$ be the norm of $α$, i.e. the determinant of the regular representaion matrix of $α$. It is well known that $N_{E/K}(α)$ $=$ $N_{L/K}(N_{E/L}(α))$ if $E/K$ is separable. I tried to find a proof of […]

$K$ is a normal subgroup of a finite group $G$ and $S$ is a Sylow $p$-subgroup of $G$. Prove that $K \cap S$ is a Sylow $p$-subgroup of $K$.

(This is my first post here, so please excuse me if I’m not following proper etiquette.) First, I noted that both $K$ and $S$ are subgroups, thus their intersection is a subgroup. We can write that $|G| = p^km$ and $|S| = p^k$ where $p$ is a prime and $m$ has no factors of $p$. […]