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For the Kronecker quiver (two vertices, two arrows in the same direction) and dimension vector (1,1), over an algebraically closed ground field, the indecomposables are naturally parameterized by poin …

What about $R = \mathbb R[x,y,z]/\langle x^2+y^2+z^2-1\rangle$ and $M$ the projective module corresponding to the tangent bundle? This seems to satisfy your criteria except probably for factorialness …

It seems unlikely (once you assume d.c.c.). Define the height of an element $x$ in $P$ to be the length of the shortest unrefinable chain from $x$ to $0$. Let $P_n$ denote the elements of $P$ whos …

There is a notion of non-commutative resolution of a singularity (in the sense of commutative algebraic geometry). See section 5 of this ICM talk by Bondal and Orlov. The following is a simple exa …

Derived categories of finite dimensional hereditary algebras for other derived categories. (Though under good circumstances other derived categories can turn out to be equivalent to derived categorie …