0
votes
1answer
66 views

Why is (gcd(f,g)) = (f,g)?

f and g are polynomials of F[X]. I can't see why (f,g) = (gcd(f,g)) ? (the ideal that f and g are the generators, and the ideal that the gcd is the generator). gcd(f,g) = a*f+b*g , for specific a ...
2
votes
0answers
72 views

Ideals (one-sided ideals) of $n×n$ upper triangular matrices

Is there any characterization of ideals (one-sided ideals) of $n\times n$ upper triangular matrices? I have just seen in monthly journal about $2 \times 2$ matrices in the below article Left and Right ...
4
votes
3answers
261 views

Definition of principal ideal

This is a pretty basic question about principal ideals - on page 197 of Katznelson's A (Terse) Introduction to Linear Algebra, it says: Assume that $\mathcal{R}$ has an identity element. For $g\in ...
4
votes
1answer
101 views

Is there a nice way to classify the ideals of the ring of lower triangular matrices?

Suppose $T$ is the subset of $M_2(\mathbb{Z})$ of lower triangular matrices, those of form $\begin{pmatrix} a & 0 \\ b & c\end{pmatrix}$. So $T$ is a subring. Now I know that the ideals of ...
3
votes
1answer
247 views

Ideals in the ring of endomorphisms of a vector space of uncountably infinite dimension.

I know that if $V$ is a vector space over a field $k,$ then $\operatorname{End}(V)$ has no non-trivial ideals if $\dim V<\infty;$ $\operatorname{End}(V)$ has exactly one non-trivial ideal if ...
1
vote
1answer
363 views

Quadratic forms and prime numbers in the sieve of Atkin

I'm studying the theorems used in the paper which explains how the sieve of Atkin works, but I cannot understand a point. For example, in the paper linked above, theorem 6.2 on page 1028 says that if ...
4
votes
2answers
243 views

Ideals of $\mathscr{I}(V) = \text{Hom}(V,V)$

I have been trying this notoriously difficult problem for quite some time but i haven't made any progress. Let $\mathscr{I}(V)$ denote the set of all homomorphisms $f : V \to V$. That is ...