Additive closed symmetric monoidal structures on R-modules. (English summary) J. Pure Appl. Algebra 215 (2011), no. 5, 789–805. Let R be a ring. Additive closed symmetric monoidal structures on the category of (left) R-modules are shown to be in correspondence with a class of left R-modules having two commuting right R-actions (called two-fold bimodules), satisfying some coherence conditions. Examples are considered in case R is one of the following: the ring of integers modulo n, a division ring, and k[Z/2Z] for k a field. In the latter case, a detailed analysis of all additive closed symmetric monoidal structures is provided. Finally, there are some structure results: the two-fold bimodule inducing the monoidal structure is shown to be faithful with respect to any R-action, and the unit of the monoidal structure is a finitely generated module with a commutative endomorphism ring. Reviewed by Adriana Bala
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