In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, especially in the area of
algebra known as
ring theory, an Ore extension, named after
Øystein Ore
Øystein Ore (7 October 1899 – 13 August 1968) was a Norwegian mathematician known for his work in ring theory, Galois connections, graph theory, and the history of mathematics.
Life
Ore graduated from the University of Oslo in 1922, with a ...
, is a special type of a
ring extension whose properties are relatively well understood. Elements of a Ore extension are called Ore polynomials.
Ore extensions appear in several natural contexts, including skew and differential
polynomial rings,
group algebras of
polycyclic groups,
universal enveloping algebras of
solvable Lie algebras, and
coordinate rings of
quantum groups.
Definition
Suppose that ''R'' is a (not necessarily
commutative)
ring,
is a
ring homomorphism, and
is a ''σ''-derivation of ''R'', which means that
is a
homomorphism of
abelian groups satisfying
:
.
Then the Ore extension
Examples
The
Weyl algebras are Ore extensions, with ''R'' any commutative
polynomial ring, ''σ'' the identity ring
endomorphism, and ''δ'' the
polynomial derivative.
Ore algebras are a class of iterated Ore extensions under suitable constraints that permit to develop a noncommutative extension of the theory of
Gröbner bases.
Properties
* An Ore extension of a
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
is a domain.
* An Ore extension of a
skew field is a non-commutative
principal ideal domain
In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, ...
.
* If ''σ'' is an
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
and ''R'' is a left
Noetherian ring then the Ore extension ''R''
hairsp;''λ''; ''σ'', ''δ'' is also left Noetherian.
Elements
An element ''f'' of an Ore ring ''R'' is called
* twosided (or invariant
), if ''R·f = f·R'', and
* central, if ''g·f = f·g'' for all ''g'' in ''R''.
Further reading
*
*
* Azeddine Ouarit (1992) Extensions de ore d'anneaux noetheriens á i.p, Comm. Algebra, 20 No 6,1819-1837. https://zbmath.org/?q=an:0754.16014
* Azeddine Ouarit (1994) A remark on the Jacobson property of PI Ore extensions. (Une remarque sur la propriété de Jacobson des extensions de Ore a I.P.) (French) Zbl 0819.16024. Arch. Math. 63, No.2, 136-139 (1994). https://zbmath.org/?q=an:00687054
*
References
{{Reflist
Ring theory