Free algebra over a ring

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