All rings are commutative with identity and all modules are unitary. In this note we give some properties of a finite collection of submodules such that the sum of any two distinct members is multiplication, generalizing those which characterize arithmetical rings. Using these properties we are able to give a concise proof of Patrick Smith’s theorem stating conditions ensuring that the sum and ...

In distributed computing systems slow working nodes, known as stragglers, can greatly extend finishing times. Coded is a technique that enables straggler-resistant computation. Most coded techniques presented to date provide robustness by ensuring the time finish depends only on set of fastest nodes. However, while stragglers do compute less work than non-stragglers, in real-world commercial cl...