A (not necessarily commutative) Krull monoid—as introduced by Wauters—is defined as a completely integrally closed monoid satisfying the ascending chain condition on divisorial two-sided ideals. We study the structure of these Krull monoids, both with ideal theoretic and with divisor theoretic methods. Among others we characterize normalizing Krull monoids by divisor theories. Based on these re...

A domain R is perinormal if every going-down overring flat and a globally localization of [2 Epstein, N., Shapiro, J. (2016). Perinormality - generalization Krull domains. Algebra 451:65–84. DOI: 10.1016/j.jalgebra.2015.12.002.[Crossref], [Web Science ®] , [Google Scholar]]. I show that global perinormality preserved in pullback construction which encompasses classical D + M construction. In do...

For a fixed prime p, let H∗(G) denote the mod p cohomology ring of a finite group G, that is, ExtFpG(Fp,Fp) interpreted algebraically and H ∗(K(G, 1); Fp) interpreted topologically. The cohomology ring is a graded commutative Noetherian ring, or equivalently, a finitely generated graded commutative Fp-algebra. Although the ring is not strictly commutative, the usual concepts from commutative ri...

In this expository note, we discuss various aspects of the theory of dimension of schemes, in particular focusing on which hypotheses are necessary in order to insure good behavior, and what sort of pathologies can otherwise arise. References are frequently exercises in disguise (or not in disguise, as the case may be). In particular, because the intended applications are geometric, we will sta...

This talk will provide a snapshot of contemporary commutative algebra. In classical commutative algebra and algebraic number theory, the Dedekind domains are the most important class of rings. Modern commutative algebra studies numerous generalizations of the Dedekind domains in attempts to generalize results of algebraic number theory. This talk will introduce a few important generalizations o...