For a non-unit a of an atomic monoid H we call $L_H\left(a\right)=k\in \mathbb{N}|a=u₁...u_{k}withirreducible{u}_i\in H$ the set of lengths of a. Let H be a Krull monoid with infinite divisor class group such that each divisor class is the sum of a bounded number of prime divisor classes of H. We investigate factorization properties of H and show that H has sets of lengths containing large gaps. Finally we apply this result to finitely generated algebras over perfect fields with infinite divisor class group.

We show that every small resolution of a 3-dimensional terminal hypersurface singularity can occur on a non-embeddable $1$-convex manifold.We give an explicit example of a non-embeddable manifold containing an irreducible exceptional rational curve with normal bundle of type $(1,-3)$. To this end we study small resolutions of $c{D}_4$-singularities.

If $A$ is a domain with the ascending chain condition on (integral) invertible ideals, then the group $I(A)$ of its invertible ideals is generated by the set $I_m(A)$ of maximal invertible ideals. In this note we study some properties of $I_m(A)$ and we prove that, if $I(A)$ is a free group on $I_m(A)$, then $A$ is a locally factorial Krull domain.

Let X be an irreducible nonsingular complex algebraic set and let K be a compact subset of X. We study algebraic properties of the ring of rational functions on X without poles in K. We give simple necessary conditions for this ring to be a regular ring or a unique factorization domain.