Facts
The following condition is equivalent to M being algebraically compact:
- For every index set I, the addition map M(I) → M can be extended to a module homomorphism MI → M (here M(I) denotes the direct sum of copies of M, one for each element of I; MI denotes the product of copies of M, one for each element of I).
Every indecomposable algebraically compact module has a local endomorphism ring.
Algebraically compact modules share many other properties with injective objects because of the following: there exists an embedding of R-Mod into a Grothendieck category G under which the algebraically compact R-modules precisely correspond to the injective objects in G.
Read more about this topic: Algebraically Compact Module
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