We show that prime Fano threefolds $Y$ of genus 8 have a multiplicative Chow–Künneth decomposition, in the sense Shen–Vial. As consequence, certain tautological subring Chow ring powers injects into cohomology.

We show that Gushel-Mukai fivefolds admit a multiplicative Chow-K\"unneth decomposition, in the sense of Shen-Vial. As consequence, certain tautological subring Chow ring powers these varieties injects into cohomology.

Let X be a topological space and R be a subring of RX. By determining some special topologies on X associated with the subring R, characterizations of maximal fixxed and maximal growing ideals in R of the form Mx(R) are given. Moreover, the classes of zR-ideals and z0R-ideals are introduced in R which are topological generalizations of z-ideals and z0-ideals of C(X), respectively. Various c...