\((r,t)\)-injectivity in the category \(\mathbf{S}\)-\(\mathbf{Act}\). (English) Zbl 1423.08005
Summary: In this paper, we show that injectivity with respect to the class \(\mathcal{D}\) of dense monomorphisms of an idempotent and weakly hereditary closure operator of an arbitrary category well-behaves. Indeed, if \(\mathcal{M}\) is a subclass of monomorphisms, \(\mathcal{M}\cap \mathcal{D}\)-injectivity well-behaves. We also introduce the notion of \((r,t)\)-injectivity in the category \(\mathbf{S}\)-\(\mathbf{Act}\), where \(r\) and \(t\) are Hoehnke radicals, and discuss whether this kind of injectivity well-behaves.
MSC:
| 08B30 | Injectives, projectives |
| 08C05 | Categories of algebras |
| 18B40 | Groupoids, semigroupoids, semigroups, groups (viewed as categories) |
| 18E40 | Torsion theories, radicals |
| 18G05 | Projectives and injectives (category-theoretic aspects) |
| 20M30 | Representation of semigroups; actions of semigroups on sets |
| 20M50 | Connections of semigroups with homological algebra and category theory |
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