Articles of abstract algebra

$\mathrm{Hom}(M,F)$ can't be determined by the underlying sets of $M,F$? where $F$ is a free module, $M$ is not.

$\mathrm{Hom}(M,F)$ can’t be determined by the underlying sets of $M,F$? where $F$ is a free module, $M$ is not a free module. The question arises from the claim: let $G : \mathbf{Mod}_R\to\mathbf{ Set}$ be the forgetful functor which assigns to each $R$- module its underlying set. Then the functor $G$ does not have a right […]

Is a contracted primary ideal the contraction of a primary ideal?

This questions is motivated by this post. Let $A\to B$ be a morphism of commutative rings and $\mathfrak q$ a contracted primary ideal in $A$ (that is, an ideal of $A$ which is contracted and primary). Question 1 Is there a primary ideal $\mathfrak r$ in $B$ whose contraction $\mathfrak r^c$ is $\mathfrak q$? Question […]

Multiplication in $\mathbb{R}^n$

I have read that it is not possible to define multiplication in $\mathbb{R}^n$ for $n\ge 3$ in any manner whatsoever so that together with usual addition it forms a field. However I have not been able to read a proof of this. Can someone give me a proof or a reference to it? Thanks

0n group that have an non-trivial element fix with each automorphism

Let $G$ be a group and $Aut(G)$ is an automorphisms groups of $G$. We know that if $Aut(G)$ is nilpotent and $G$ is not cyclic of odd order, then $G$ has an non-trivial element such that fix by all automorphisms. Also it is clear that if $G$ has a characteristic subgroup of order 2, then […]

Why $\mathbf Z/(2, 1+\sqrt{-5})\simeq \mathbf Z/(2,x+1,x^2+5)$?

Why is $(2, 1+\sqrt{-5})$ not principal? \begin{align*}\mathbf Z[\sqrt{-5}]/(2, 1+\sqrt{-5})&\simeq \mathbf Z[x]/(2,x+1,x^2+5)\simeq \mathbf Z_2[x]/(x+1,x^2+1)\\ &=\mathbf Z_2[x]/\bigl(x+1,(x+1)^2\bigr)=\mathbf Z_2[x]/(x+1)\simeq\mathbf Z_2. \end{align*} It is said to be used that $(R/I)/(J/I)\simeq R/J$, but can you explain in more detail? And I still don’t understand about the other homomorphisms and equations (especially $\mathbf Z_2[x]/(x+1,x^2+1)=\mathbf Z_2[x]/\bigl(x+1,(x+1)^2)$

In finite semigroup, if $ef = f$ for $e,f \in E(S)$ and $e = xy, f = yx, s = sx = se = sf$ then we could not have $fe = e$

Let $S$ be a finite semigroup. For two idempotents $e, f$ we set $$ f \le e :\Leftrightarrow ef = f $$ Now let $s \in S$ and $e,f \in S$ be two idempotents. Suppose $s = se = sf$ and there exists $x, y\in S$ such that $$ s = sx, e = xy, […]

quivers and tensor product

I am studyng this lemma of Assem: Good, now let $Q$ the kronecker quiver : then there is algebra isomorphism $KQ\cong \begin{bmatrix} K &0 \\ K^2 & K \end{bmatrix}$ where $K^2$ is considered as a K-K-bimódule in the way $a(x,y)=(ax,ay)$, $(x,y)b=(xb,yb)$ for all $a,b,x,y$ in $K$. I would like to understand what it is $\varphi […]

Proofs with steps of division and setting things not equal to zero

I am self studying from A Book of Abstract Algebra by Charles C. Pinter. In chapter 3 problem set B problem 4 it asks to show whether $(a,b)\star(c,d)=(ac-bd,ad+bc)$ on the set $ \{(x,y) \in \mathbb R^2 | (x,y) \not= (0,0)\} $ is a group. Currently I am in the middle of showing that there is […]

Find an integral domain $D$ containing an irreducible element $p$ such that $D/\langle p \rangle$ is not a field.

Find an integral domain $D$ containing an irreducible element $p$ such that $D/\langle p \rangle$ is not a field. I’m working on homework. I think I need to find p such that the ideal generated by $p$ is not maximal. So I think I need an integral domain which is not a PID. If $p$ […]

Show that there is such a Sylow subgroup

I want to show that if $G$ is finite and $f:G\rightarrow H$ is a group epimorphism and if $Q\in \text{Syl}_p(H)$ then there is a $P\in \text{Syl}_p(G)$ with $Q=f(P)$. $$$$ I have done the following: We have that $f:G\rightarrow H$ is a group epimorphism, so $f$ is surjective. That means that for every $y\in H$ there […]