# Free algebra over a ring

From Encyclopedia of Mathematics

* that is associative and commutative*

A free algebra in the variety of algebras over (see Rings and algebras). The elements of such a free algebra with a free generating system are linear combinations of elements of the free groupoid generated by with coefficients from . In other words, this free algebra is a free module over with the above-mentioned groupoid as its base. When is the ring of integers, a free algebra over is called a free ring (cf. Free associative algebra). A non-null subalgebra of a free algebra over a field is a free algebra.

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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