When are induction and coinduction functors isomorphic?
When is the category of flat modules abelian?
Let Fl(R) denote the category of flat right modules over an associative ring R. We find necessary and sufficient conditions for Fl(R) to be a Grothendieck category, in terms of properties of the ring R.
When is a cogenerator in a topos ?
When projective does not imply flat, and other homological anomalies.
When spectra of lattices of -ideals are Stone-Čech compactifications
Let be a completely regular Hausdorff space and, as usual, let denote the ring of real-valued continuous functions on . The lattice of -ideals of has been shown by Martínez and Zenk (2005) to be a frame. We show that the spectrum of this lattice is (homeomorphic to) precisely when is a -space. This we actually show to be true not only in spaces, but in locales as well. Recall that an ideal of a commutative ring is called a -ideal if whenever two elements have the same annihilator and...
When the product-preserving functors preserve limits
Words, free algebras, and coequalizers