Free algebra over a ring

$\Phi$ that is associative and commutative

A free algebra in the variety of algebras over $\Phi$ (see Rings and algebras). The elements of such a free algebra with a free generating system $X$ are linear combinations of elements of the free groupoid generated by $X$ with coefficients from $\Phi$. In other words, this free algebra is a free module over $\Phi$ with the above-mentioned groupoid as its base. When $\Phi$ is the ring of integers, a free algebra over $\Phi$ is called a free ring (cf. Free associative algebra). A non-null subalgebra of a free algebra over a field $\Phi$ is a free algebra.