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Difference between revisions of "Free algebra over a ring"

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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041500/f0415001.png" /> that is associative and commutative''
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''$\Phi$ that is associative and commutative''
  
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A free algebra in the variety of algebras over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041500/f0415002.png" /> (see [[Rings and algebras|Rings and algebras]]). The elements of such a free algebra with a free generating system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041500/f0415003.png" /> are linear combinations of elements of the [[Free groupoid|free groupoid]] generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041500/f0415004.png" /> with coefficients from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041500/f0415005.png" />. In other words, this free algebra is a [[Free module|free module]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041500/f0415006.png" /> with the above-mentioned groupoid as its base. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041500/f0415007.png" /> is the ring of integers, a free algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041500/f0415008.png" /> is called a free ring (cf. [[Free associative algebra|Free associative algebra]]). A non-null subalgebra of a free algebra over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041500/f0415009.png" /> is a free algebra.
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A free algebra in the variety of algebras over $\Phi$ (see [[Rings and algebras|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|free groupoid]] generated by $X$ with coefficients from $\Phi$. In other words, this free algebra is a [[Free module|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|Free associative algebra]]). A non-null subalgebra of a free algebra over a field $\Phi$ is a free algebra.

Revision as of 07:05, 3 October 2014

$\Phi$ 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.

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=14483
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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