satisfies the Leibniz rule; i.e., . Given a Poisson algebra , consider the vector space of formal power series over . If has a structure of an associative algebra with multiplication such that, for , , then is called a deformation quantization of . A quantized enveloping algebra. The dual of such an algebra is, philosophically speaking, the coordinate ring of a quantum group (in one of the definitions). Constructions Subalgebras A subalgebra of an R-algebra A is a subset of A which is both a subring and a submodule of A. That is, it must be closed under addition, ring multiplication, scalar multiplication, and it must contain the identity element of A. Quotient algebras Let A be an R-algebra. Any ring-theoretic ideal I in A is automatically an R-module since r · x = (r1A)x. This gives the quotient ring A / I the structure of an R-module and, in fact, an R-algebra. It follows that any ring homomorphic image of A is also an R-algebra. Direct products The direct product of a family of R-algebras is the ring-theoretic direct product. This becomes an R-algebra with the obvious scalar multiplication. Free products One can form a free product of R-algebras in a manner similar to the free product of groups. The free product is the coproduct in the category of R-algebras. Tensor products The tensor product of two R-algebras is also an R-algebra in a natural way. See tensor product of algebras for more details. Given a commutative ring R and any ring A the tensor product R ⊗Z A can be given the structure of an R-algebra by defining r · (s ⊗ a) = (rs ⊗ a). The functor which sends A to R ⊗Z A is left adjoint to the functor which sends an R-algebra to its underlying ring (forgetting the module structure). See also: Change of rings. Separable algebra Let A be an algebra over a commutative ring R. Then the algebra A is a right module over with the action . Then, by definition, A is said to separable if the multiplication map splits as an -linear map, where is an -module by . Equivalently, is separable if it is a projective module over ; thus, the -projective dimension of A, sometimes called the bidimension of A, measures the failure of separability. Finite-dimensional algebra Let A be a finite-dimensional algebra over a field k.