Articles of abstract algebra

An example about finitely cogenerated modules

We know the fact that: If a module $M$ is finitely cogenerated, then every module that cogenerates $M$ finitely cogenerates $M$. Conversely, it is not true. I find an example in the book “Rings and Categories of modules” written by Frank W. Anderson and Kent R. Fuller. Example The abelian group $\bigoplus_{\mathbb{P}}{\mathbb{Z}_p}$, ($p$ is a […]

Trigonometric solution to solvable equations

The algebraic equations in one variable, in the general case, cannot be solved by radicals. While the basic operations and root extraction applied to the coefficients of the equations of degree $2, 3, 4 $ are sufficient to express the general solution, for arbitrary degree equations, the general solution can be expressed only in terms of much […]

True/False about ring and integral domain

I have some true or false questions and would like to have your help to check on it. A). in a ring R, if $x^2=x$, $\forall x\in R$, then R is commutative For (A), when looking at $(x+y)^2$, it has $x+y=(x+y)^2=x^2+xy+yx+y^2$ and then yx+xy=0, and from 2x=4x, therefore 2x=0. how this play a role here? […]

Determining the presentation matrix for a module

I am trying to study some module theory using the book “Algebra” by Michael Artin (2nd Edition, to be precise), and I can’t really fathom what is written in Section 14.5. Left multiplication by an $m \times n$ matrix defines a homomorphism of $R$-modules $A: R^n \rightarrow R^m$. Its image consists of all linear combinations […]

Decomposition Theorem for Posets

There is in module theory the following (krull-shmidt) theorem: If $(M_i)_{i\in I}$ and $(N_j)_{j\in J}$ are two families of simple module such that $\bigoplus\limits_{i\in I} M_i \simeq \bigoplus\limits_{j\in J} M_j$, then there exists a bijection $\sigma:I\rightarrow J$ such that $M_i \simeq N_{\sigma(i)}$. Is there a similar kind of theorem for partially ordered sets? More precisely, […]

Proof that $\mathbb{R}$ is not a finite dimensional vector space

How can we prove that the vector space of polynomials in one variable, $\mathbb{R}[x]$ is not finite dimensional?

Finding the kernel of ring homomorphisms from rings of multivariate polynomials

I am trying to find the kernels of the following ring homomorphisms: $$ f:\Bbb C[x,y]\rightarrow\Bbb C[t];\ f(a)=a\ (a\in\Bbb C),f(x)=t^2,f(y)=t^5. $$ $$ g:\Bbb C[x,y,z]\rightarrow\Bbb C[t,s];\ g(a) = a\ (a\in\Bbb C), g(x)=t^2,g(y)=ts,g(z)=s^2. $$ $$ h:\Bbb C[x,y,z]\rightarrow\Bbb C[t];\ h(a)=a\ (a\in\Bbb C), h(x)=t^2, h(y)=t^3, h(z)=t^4. $$ I want to write them as ideals generated by as few elements as […]

Weyl group, permutation group

Let $U(n)$ be the unitary group and $T$ its maximal torus (group of diagonal matrix) and $N(T)$ the normalizer of $T$ in $G$. Why $N(T)/T$ is the permutation group $S_{n}$?

Has $S$ infinitely many nilpotent elements?

Let $S$ be a ring with identity (but not necessarily commutative) and $f:M_{2}(\mathbb R)→S$ a non zero ring homomorphism ($M_{2}(\mathbb R)$ is the ring of all $2\times 2$ matrices). Has $S$ infinitely many nilpotent elements?

The degree of the extension $F(a,b)$, if the degrees of $F(a)$ and $F(b)$ are relatively primes.

Let $E$ be an extension of $F$, and let $a, b \in E$ be algebraic over $F$. Suppose that the extensions $F(a)$ and $F(b)$ of $F$ are of degrees $m$ and $n$, respectively, where $(m,n)=1$. Show that $[F(a,b):F]=mn$. Since $[F(a,b):F]=[F(a,b):F(a)][F(a):F]$ and $[F(a):F]=n$ we have $n|[F(a,b):F]$ with the same argument we prove that $m|[F(a,b):F]$, then $mn|[F(a,b):F]$ […]