Articles of abstract algebra

Is every regular element of a ring invertible?

When I am reading a paper, I found the definition of a new ring as following: In this definition, if every central regular element is invertible, i.e., how to understand the invertible element of u? how to prove it is a ring?Moreover, is the product of a regular element and a unit also a unit? […]

In a P.I.D., if $a^m = b^m$ and $a^n = b^n$ for $m, n \in \mathbb{N}$ with $\gcd(m,n) = 1$, then $a=b$

Let $R$ be a principal ideal domain, and $a, b \in R$ with $a^m = b^m$ and $a^n = b^n$ for $m, n \in \mathbb{N}$, and $\gcd(m, n) = 1$. I now want to show that we then already have $a = b$. I think the statement is easily seen to be true if $a […]

Field extensions and monomorphisms

I am working through some algebra exercises and got stuck with the following problem: I am working in the finite field $\mathbb{Z}_5$. Let $f \in \mathbb{Z}_5[x]$ and $f(x) = x^2 + 2$. By simple trial and error, I checked that this polynomial is irreducible, hence $\mathbb{Z}_5/(f) \cong GF(5^2)$, denoting the Galois field ($f$ is of […]

Isomorphism and cyclic modules

Prove that $M$ is a cyclic $R$-module if and only if exists a left ideal $I\subset R$ such that $M \simeq R/I$. I’m not sure how to even start this proof. I’ve been told that I could use an annhilator set to prove this, but I don’t see the relation.

$ \mathbb Z$ is not isomorphic to any proper subring of itself.

Show that the ring $ \mathbb Z$ is not isomorphic to any proper subring of itself. Is the cardinality main reason for not being isomorphic?? Please Help!!

How to find galois group? E.g. $\mathrm{Gal}(\mathbb Q(\sqrt 2, \sqrt 3 , \sqrt 5 )/\mathbb Q$

I want to know how to find a Galois group. I know I need to look at automorphisms but I am not sure of the method. Could someone give me an idea, e.g. with the example: $\mathrm{Gal}(\mathbb Q(\sqrt 2,\sqrt3,\sqrt5)/\mathbb Q$ Thanks

If $G$ is isomorphic to all non-trivial cyclic subgroups, prove that $G\cong \mathbb{Z}$ or $G\cong \mathbb{Z}_p$

Assuming $G$ is isomorphic to all its non-trivial subgroups, prove that $G$ is isomoprhic to either $\mathbb Z$ or $\mathbb Z_p$ for $p$ prime.

How can I find the kernel of $\phi$?

We have the homomorhism $\phi: \mathbb{C}[x,y] \to \mathbb{C}$ with $\phi(z)=z, \forall z \in \mathbb{C}, \phi(x)=1, \phi(y)=0$. I have shown that for $p(x,y)=a_0+\sum_{k,\lambda=1}^m a_{k \lambda} (x-1)^k y^{\lambda}$, we have $\phi(p(x,y))=a_0 \in \mathbb{C}$. How can I find the kernel of $\phi$ ?

Representing Elementary Functions in a CAS

I’ve looked through several books about computer algebra. They are surprisingly scarce about how to actually represent elementary functions. Basically, as far as I understood elementary functions are roughly $$\textbf{Quot}\left(\mathbb K[x, \exp x, \exp x^2 \log x, \ldots] \right)$$ that is rational functions over field $\mathbb K$, with variables (or kernels) $x$, $\exp x$, $\exp […]

To show that either $R$ is a field or $R$ is a finite ring with prime number of elements and $ab = 0$ for all $a,b \in R$.

Let $R$ be a commutative ring such that $R$ has no nontrivial ideal. Then show that either R is a field or R is a finite ring with prime number of elements and $ab = 0$ for all $a,b \in R$. I am facing difficulty in proving the above!!