Graded Algebra and Polynomial Rings
Polynomial rings and their quotients by homogeneous ideals are typical graded algebras. Conversely, if S is a graded algebra generated over the field K by n homogeneous elements g1, ..., gn of degree 1, then the map which sends Xi onto gi defines an homomorphism of graded rings from onto S. Its kernel is a homogeneous ideal I and this defines an isomorphism of graded algebra between and S.
Thus, the graded algebras generated by elements of degree 1 are exactly, up to an isomorphism, the quotients of polynomial rings by homogeneous ideals. Therefore, the remainder of this article will be restricted to the quotients of polynomial rings by ideals.
Read more about this topic: Hilbert Series And Hilbert Polynomial
Famous quotes containing the words graded, algebra and/or rings:
“I dont want to be graded on a curve.”
—Mary Carillo (b. 1957)
“Poetry has become the higher algebra of metaphors.”
—José Ortega Y Gasset (18831955)
“Ah, Christ, I love you rings to the wild sky
And I must think a little of the past:
When I was ten I told a stinking lie
That got a black boy whipped....”
—Allen Tate (18991979)