In this paper, we are concerned with certain invariants of modules, called reducing invariants, which have been recently introduced and studied by Araya–Celikbas and Araya–Takahashi. We raise the question whether the residue field of each commutative Noetherian local ring has finite reducing projective dimension and obtain an affirmative answer for the question for a large class of local rings. Furthermore, we construct new examples of modules of infinite projective dimension that have finite reducing projective dimension and study several fundamental properties of reducing dimensions, especially properties under local homomorphisms of local rings.

We study the growth of p-primary Selmer groups of abelian varieties with good ordinary reduction at p in ${{Z}}_p$ -extensions of a fixed number field K. Proving that in many situations the knowledge of the Selmer groups in a sufficiently large number of finite layers of a ${{Z}}_p$ -extension over K suffices for bounding the over-all growth, we relate the Iwasawa invariants of Selmer groups in different ${{Z}}_p$ -extensions of K. As applications, we bound the growth of Mordell–Weil ranks and the growth of Tate-Shafarevich groups. Finally, we derive an analogous result on the growth of fine Selmer groups.

We prove that each positive power of the maximal ideal of a commutative Noetherian local ring is Tor-rigid and strongly rigid. This gives new characterizations of regularity and, in particular, shows that such ideals satisfy the torsion condition of a long-standing conjecture of Huneke and Wiegand.

This paper concerns a generalization of the Rees algebra of ideals due to Eisenbud, Huneke and Ulrich that works for any finitely generated module over a noetherian ring. Their definition is in terms of maps to free modules. We give an intrinsic definition using divided powers.

If R is a Dedekind domain, then div splits i.e.; the maximal divisible submodule of every R-module M is a direct summand of M. We investigate the status of this result for some finite-dimensional hereditary algebras. We use a torsion theory which permits the existence of torsion-free divisible modules for such algebras. Using this torsion theory we prove that the algebras obtained from extended Coxeter- Dynkin diagrams are the only such hereditary algebras for which div splits. The field of rational functions plays an essential role. The paper concludes with a new type of infinite-dimensional indecomposable module over a finite-dimensional wild hereditary algebra.

Let R be a commutative ring with identity. An R-module M is said to be distributive if the lattice of submodules of M is distributive. We characterize such modules and study their properties.