\(t\)-closedness (Q2752916)

A commutative ring \(R\) with total quotient ring \(K\) is said to be \(t\)-closed if whenever \(b^2-rb\), \(b^3-rb^2\in R\) for \(r\in R\) and \(b\in K\), then \(b\in R\). A \(t\)-closed ring is seminormal and an integrally closed ring is \(t\)-closed. This notion was introduced and studied in a series of papers in the mid 1980's by Koyama, Onoda, Sugatani, and Yoshida because of its relationship to quasinormality (recall that \(R\) is called quasinormal if \(\text{Pic} (R[X, X^{-1}])= \text{Pic} (R))\). For example, a one-dimensional Noetherian integral domain is quasinormal if and only if it is \(t\)-closed. In the spirit of Swan's treatment of seminormality [\textit{R. G. Swan}, J. Algebra 67, 210-229 (1980; Zbl 0473.13001)], the authors have defined a commutative ring \(R\) to be \(t\)-closed if whenever \(x^3+rxy-y^3=0\) for \(x,y,r\in R\), then \(x=z^2-rz\) and \(y=z^3-rz^2\) for some \(z\in R\) (this agrees with the earlier definition if \(K\) is absolutely flat). NEWLINENEWLINENEWLINEIn this survey article, the authors review many aspects of \(t\)-closedness (most of them were developed by the authors in a series of papers in the 1990's). Some of the topics covered include extensions of \(t\)-closed rings, \(u\)-closed rings, \(t\)-closed pairs, and strongly \(t\)-closed rings.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00012].