Closure operations and actions of algebraic groups

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When a group scheme G acts on a commutative rig, we consider the problem, if the result of closure operations on G-ideals again a G-ideal. We get some results in special cases. We also studied basics on equivariant local cohomology, and obtained the equivariant versions of Matlis and the local duality. We also defined the equivariant versions of prime and primary ideals, and obtained the equivariant versions of the existence and the uniqueness of the primary decomposition. We also succeeded in proving the Matijevic-Roberts type theorems on singularities in positive characteristic.