View source for Free algebra over a ring

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''$\Phi$ 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 magma]] generated by $X$ with coefficients from $\Phi$. In other words, this free algebra is a [[free module]] over $\Phi$ with the above-mentioned magma as its base. When $\Phi$ is the ring of integers, a free algebra over $\Phi$ is called a free ring (cf. [[Free associative algebra]]).  Kurosh showed that a non-null subalgebra of a free algebra over a field $\Phi$ is a free algebra.

====References====
* Kurosh, A. "Nonassociative free algebras and free products of algebras".  Mat. Sb., N. Ser. '''20(62)''' (1947) 239-262  {{ZBL|0041.16803}} (In Russian with English summary)
* Mikhalev, Alexander A.; Shpilrain, Vladimir; Yu, Jie-Tai, ''Combinatorial methods. Free groups, polynomials, and free algebras'', CMS Books in Mathematics '''19''' Springer (2004) ISBN 0-387-40562-3 {{ZBL|1039.16024}}