Add(M) lies in Prod(M) exactly for Sigma-pure-injective objects under a non-omega-measurable cardinality hypothesis, and Prod(M)=Add(M) exactly for Sigma-pure-injective product-rigid objects, in two broad categorical settings.
Angeleri-H¨ ugel, J
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Comparing $\mathrm{Add}(M)$ with $\mathrm{Prod}(M)$
Add(M) lies in Prod(M) exactly for Sigma-pure-injective objects under a non-omega-measurable cardinality hypothesis, and Prod(M)=Add(M) exactly for Sigma-pure-injective product-rigid objects, in two broad categorical settings.