Let $A$ be a noetherian domain, $\mathfrak{m}$ а maximal ideal, $s$ a non-zero element of $\mathfrak{m}$, $d= \dim A_\mathfrak{m}$. Is the following claim true? Claim: For any $\e …

Let $A$ be a local ring over $\mathbb{C}$, which moreover is a finite dimensional $\mathbb{C}$-vector space. When is $A$ a subring of $\mathbb{C}[t]/t^n$? What …

As in title. Are there only finite many maximal left ideals for a left Artinian ring?

Given a commutative Artin algebra $A$ over an algebraically closed field $k$ one has a decomposition $A=A_1\oplus\ldots\oplus A_n$ into local Artin subalgebras, see for example Ati …

I am having trouble showing that a ring which is left Artinian and right Noetherian is right Artinian.

I wonder if the triangulated category K(R-Mod) is compactly generated when R is an artin algebra? R-Mod denotes all left R-modules. I understand this would be true if R has finite …

Let f:X -> Y be a morphism of schemes over a field k. Can one check that f is formally smooth using only Artin rings of the form k'[t]/t^n, where k' is also a field? Considering c …