Articles of ring theory

Why characters are continuous

According to Wikipedia: ”Every character is automatically continuous from $A$  to $\mathbb C$, since the kernel of a character is a maximal ideal, which is closed. ” where $A$ is a Banach algebra. How does it follow that a character is continuous? Is the following a theorem? A ring homomorphism $f: R \to S$ is […]

For a field $K$, is there a way to prove that $K$ is a PID without mentioning Euclidean domain?

I know that if $K$ is a field then $K[x]$ is a Euclidean domain and every Euclidean domain is a PID. In this way I can prove that $K[x]$ is a PID. But is there a method to show $K[x]$ is a PID directly from the definition? I mean a usual procedure is to design […]

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 […]

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?

Showing that the only units in $\mathbb{Z}$ are $1,\, -1, \, i, \, -i$?

How would you show that the only units in $\mathbb{Z}[i] :=\{a + ib \, |\, a,b\in \mathbb{Z}\}$ are $1,\, -1, \, i, \, -i$?

To find all odd integers $n>1$ such that $2n \choose r$ , where $1 \le r \le n$ , is odd only for $r=2$

For which odd integers $n>1$ is it true that $2n \choose r$ where $1 \le r \le n$ is odd only for $r=2$ ? I know that $2n \choose 2$ is odd if $n$ is odd but I want to find those odd $n$ for which the only value of $r$ between $1$ and $n$ […]

$S^{-1}A \cong A/(1-ax)$

This question already has an answer here: Localisation isomorphic to a quotient of polynomial ring [duplicate] 4 answers

Example of a compact module which is not finitely generated

Let $R$ be a ring and $M$ be an $R$-module. Definition: $M$ is called compact if $\text{Hom}_R(M,-)$ commutes with direct sums, that is, if for any set $I$ and any $I$-indexed family of $R$-modules $\{N_i\}_{i\in I}$ the canonical map of abelian groups $$\bigoplus\limits_{i\in I}\text{Hom}_R\left(M,N_i\right)\longrightarrow\text{Hom}_R\left(M,\bigoplus\limits_{i\in I} N_i\right)$$ is an isomorphism. Example: Any finitely generated $R$-module is […]

Can this quick way of showing that $K/(Y-X^2)\cong K$ be turned into a valid argument?

I’ve been trying to show that $$ K[X,Y]/(Y-X^2)\cong K[X] $$ where $K$ is a field, $K[X]$ and $K[X,Y]$ are the obvious polynomial rings over the indeterminates $X$ and $Y$ and $(Y-X^2)$ is the ideal generated by the polynomial $Y-X^2$. Though I’m sure there’s a fairly easy way to find an explicit isomorphism between the two […]

dimension of tensor products over a submodule

Let $k$ be a field, $A$ be a finite dimensional $k$-algebra (say of dimension $n_A$) and let $B\subset A$ be a sub-algebra. What can be said about the $k$-dimension of $A \otimes_B A$ ? The easiest case is when $A$ is $B$-free of rank $r_A$. Then $A \otimes_B A$ has $B$-rank $r_A^2$, and $k$-dimension $n_B^{r_A^2}$ […]