In this paper we look at the properties of modules and prime ideals in finite dimensional noetherian rings. This paper is divided into four sections. The first section deals with noetherian one-dimensional rings. Section Two deals with what we define a “zero minimum rings” and explores necessary and sufficient conditions for the property to hold. In Section Three, we come to the minimal prime i...

For any commutative ring $A$ we introduce a generalization of $S$--noetherian rings using here\-ditary torsion theory $\sigma$ instead multiplicatively closed subset $S\subseteq{A}$. It is proved that totally noetherian w.r.t. local property, and if w.r.t $\sigma$, then finite type.

In this note we improve and extend duality theorems for crossed products obtained by M. Koppinen (C. Chen) from the case of base fields (Dedekind domains) to the case of an arbitrary Noetherian commutative ground rings under fairly weak conditions. In particular we extend an improved version of the celebrated Blattner-Montgomery duality theorem to the case of arbitrary Noetherian ground rings.

Intersection sheaves are usually defined for a proper flat surjective morphism of Noetherian schemes of relative dimension d and for d + 1 invertible sheaves on the ambient scheme. In this article, the construction is generalized to the equidimensional proper surjective morphisms over normal separated Noetherian schemes. Applications to the studies on family of effective algebraic cycles and on...

We study row-finite Leavitt path algebras. We characterize the row-finite graphs E for which the Leavitt path algebra is weakly Noetherian. Our main result is that a Leavitt path algebra is weakly Noetherian if and only if there is ascending chain condition on the hereditary and saturated closures of the subsets of the vertices of the graph E.

In this paper we classify noetherian hereditary abelian categories satisfying Serre duality in the sense of Bondal and Kapranov. As a consequence we obtain a classification of saturated noetherian hereditary categories. As a side result we show that when our hereditary categories have no nonzero projectives or injectives, then the Serre duality property is equivalent to the existence of almost ...