Another useful example of a monad is the symmetric algebra functor on the category of formula_103-modules for a commutative ring formula_103.formula_105sending an formula_103-module formula_107 to the direct sum of symmetric tensor powersformula_108where formula_109. For example, formula_110 where the formula_103-algebra on the right is considered as a module. Then, an algebra over this monad are commutative formula_103-algebras. There are also algebras over the monads for the alternating tensors formula_113 and total tensor functors formula_114 giving anti-symmetric formula_103-algebras, and free formula_103-algebras, soformula_117where the first ring is the free anti-symmetric algebra over formula_103 in formula_119-generators and the second ring is the free algebra over formula_103 in formula_119-generators.