An associative unital algebra over a commutative unital ring is a monoid internal to the category of modules over that ring with respect to the usual tensor product of such modules.
Let be a commutative unital ring. All -modules considered are supposed to be unital. Let be a tensor product functor for -modules and let be the corresponding associator and and the left and right unitors.
An associative unital algebra over is any triple such that
and such that the following conditions are met: