# Free algebra over a ring

From Encyclopedia of Mathematics

2020 Mathematics Subject Classification: *Primary:* 17A50 [MSN][ZBL]

*$\Phi$ 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

**How to Cite This Entry:**

Free algebra over a ring.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Free_algebra_over_a_ring&oldid=54369

This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article