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Since rings may be regarded as Z-algebras, a free ring on E can be defined as the free algebra Z E . Over a field, the free algebra on n indeterminates can be constructed as the tensor algebra on an n-dimensional vector space. For a more general coefficient ring, the same construction works if we take the free module on n generators.
Every module over a division ring is a free module (has a basis); consequently, much of linear algebra can be carried out over a division ring instead of a field. The study of conjugacy classes figures prominently in the classical theory of division rings; see, for example, the Cartan–Brauer–Hua theorem.
This defines a functor Ring → CRing which is left adjoint to the inclusion functor, so that CRing is a reflective subcategory of Ring. The free commutative ring on a set of generators E is the polynomial ring Z[E] whose variables are taken from E. This gives a left adjoint functor to the forgetful functor from CRing to Set.
The functor which sends A to R ⊗ Z A is left adjoint to the functor which sends an R-algebra to its underlying ring (forgetting the module structure). See also: Change of rings. Free algebra A free algebra is an algebra generated by symbols. If one imposes commutativity; i.e., take the quotient by commutators, then one gets a polynomial algebra.
An example of a simple ring that is not semisimple is the Weyl algebra. The Weyl algebra also gives an example of a simple algebra that is not a matrix algebra over a division algebra over its center: the Weyl algebra is infinite-dimensional, so Wedderburn's theorem does not apply.
In ring theory, a branch of mathematics, a ring is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, x 2 = 0 implies x = 0. A commutative algebra over a commutative ring is called a reduced algebra if its underlying ring is reduced.
A ring in which all idempotents are central is called an abelian ring. Such rings need not be commutative. A ring is directly irreducible if and only if 0 and 1 are the only central idempotents. A ring R can be written as e 1 R ⊕ e 2 R ⊕ ... ⊕ e n R with each e i a local idempotent if and only if R is a semiperfect ring.
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the given group. As a ring, its addition law is that of the free module and its multiplication extends "by ...