T-Closedness

Abstract

We first give some informations in order to situate t-closedness in the context of literature. The notion of t-closedness originates in K-theory and more specifically in the study of Picard groups Pic(R[X]) and Pic (R [X, X ^-1]) where R is a commutative ring. Extending works of Traverso [34], Gilmer, Heitmann [9] and others, Rush gave a characterization for a reduced ring R with a finite minimal spectrum to verify Pic(R) = Pic(R[X]) [29]. To be more explicit, Pic(R) = Pic(R[X]) if and only if R is seminormal in the total quotient ring Tot(R) of R, that is to say if a ∈ Tot(R) and a ², a ³ ∈ R then a ∈ R. Later on, Swan succeeded in giving a general theory by introducing the notion of seminormal ring [33]. A ring R is called seminormal if whenever b, c ∈ R satisfy b ³ = c ², there is some t ∈ R such that b = t ², c = t ³ . A seminormal ring is reduced as Costa showed. Then if R is reduced, Pic(R) = Pic(R[X]) holds if and only if R is seminormal. Now a ring R is called quasinormal if Pic(R) = Pic (R [X, X ^-1]). This notion was studied by Bass-Murthy [4], Greco [10] and Rush [30]. It is natural to ask whether quasinormality of a ring can be characterized by a condition similar to seminormality. To this end, Asanuma introduced u-closedness for rings [3]. A ring R is u-closed when a ∈ Tot(R) and a ² — a, a ³ — a ² ∈ R imply a ∈ R. Asanuma showed that a one-dimensional noetherian integral domain R is quasinormal if and only if R is seminormal and u-closed.