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.
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
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