Difference between revisions of "Free algebra over a ring"
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''$\Phi$ that is associative and commutative'' | ''$\Phi$ 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]]). | + | 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==== | ====References==== | ||
| − | * 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}} | + | * 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}} | ||
Latest revision as of 08:20, 12 November 2023
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=37042
Free algebra over a ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Free_algebra_over_a_ring&oldid=37042
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article