Articles of ring theory

Constructing a quotient ring in GAP using structure constants

I need to construct the following ring in GAP: $$Z_4(u) / \langle u^2-2u=0 \rangle. $$ This is what I tried and it didn’t work: gap> R:=PolynomialRing(Integers mod 4,”u”);AssignGeneratorVariables(R); <monoid>[u] #I Global variable `u’ is already defined and will be overwritten #I Assigned the global variables [ u ] gap> I:=Ideal(R,[u^2-2*u]); <two-sided ideal in <monoid>[u], (1 […]

Is $\mathbb{Z}\over \langle x+3\rangle$ field?

Is $\mathbb{Z}[x]\over \langle x+3\rangle $ field? Can I say $x+3$ is irreducible over $\mathbb{Z}$, so it is field.

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? […]

$ \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 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 $\phi$ ?

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!!

Kernel of homomorphism $(J:I)\rightarrow \text{Hom}_R(R/I, R/J)$

We wanted to do this: Let $R$ be a commutative ring and $I,J$ ideals of $R$. Then we have the $R$-isomorphism (of modules): $$\text{Hom}_R(R/I, R/J)\cong (J:I)/J$$ where $$(J:I)=\{x\in R\mid Ix\subseteq J\}\supseteq J$$ is the quotient of the ideals $J,I$. We defined the homomorphism: $$h:(J:I)\rightarrow \text{Hom}_R(R/I, R/J)$$ $$x\mapsto h_x$$ where $$h_x:R/I\rightarrow R/J$$ $$r+I\mapsto xr+J$$ We proved […]

Show that, if $ a + bi$ is prime in $\mathbb{Z} $, then $a – bi$ is prime in $\mathbb{Z}$

Show that, if $ a + bi$ is prime in $\mathbb{Z} [i]$, then $a – bi $ is prime in $\mathbb{Z}[i]$ Since $\mathbb{Z}[i]$ is $ED$, then if $a+bi$ is irreducible then $a+bi$ is prime. But now how I can relate $a+bi$ with $a-bi$? Thanks for help.

Isn't $x^2+1 $ irreducible in $\mathbb Z$, then why is $\langle x^2+1 \rangle$ not a maximal ideal in $\mathbb Z?$

Isn’t $ x^2+1$ irreducible in $\mathbb Z$, then why is $\langle x^2+1 \rangle$ not a maximal ideal in $\mathbb Z[x]$? $ x^2+1$ cannot be broken down further non trivially in $\mathbb Z[x]$. hence, it’s irreducible in $\mathbb Z[x]$. Hence, shouldn’t $\mathbb Z[x]/\langle x^2+1 \rangle$ be a field and hence, $ x^2+1$ a maximal ideal in […]

Transcendence degree of $K$

Let $K$ be field. How do I proof that transcendence degree of $K[X_1,X_2,\ldots,X_n]$ is $n$? The set $\{X_1,X_2,\ldots,X_n\}$ is algebraically independent over $K$. So, I have to show that every subset of size greater than $n$ is algebraically dependent.