5
votes
0answers
156 views

Is there a simple example of a ring that satifies the DCC on two-sided ideals, but doesn't satisfy the ACC on two-sided ideals?

It follows from the Hopkinsā€“Levitzki theorem that if a ring satisfies the DCC on left ideals, then it also satisfies the ACC on left ideals. I've been trying to find a counterexample to the following ...
6
votes
0answers
126 views

Is there an upper bound to the number of rings that can be obtained from a semigroup with zero by defining an additive operation?

Let $\mathscr S$ be the class of all semigroups with zero. For $(S,\times,0)\in\mathscr S,$ I want to count additive operations $+$ on $S$ such that $(S,+,\times,0)$ is a ring (possibly without ...
3
votes
1answer
216 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 ...