commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...

, a Rees decomposition is a way of writing a

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

in terms of polynomial subrings. They were introduced by .

Definition

Suppose that a ring ''R'' is a quotient of a polynomial ring ''k'' 'x''₁,...over a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

by some homogeneous ideal. A Rees decomposition of ''R'' is a representation of ''R'' as a direct sum (of vector spaces) :

R = \bigoplus_\alpha \eta_\alpha k theta_1,\ldots,\theta_

where each ''η''_''α'' is a homogeneous element and the ''d'' elements ''θ''_''i'' are a homogeneous system of parameters for ''R'' and ''η''_''α''''k'' 'θ''_{''f''_''α''+1},...,''θ''_''d''⊆ ''k'' 'θ''₁, ''θ''_{''f''_''α''}

References

* * Commutative algebra {{commutative-algebra-stub

Definition

See also

References