| 11. | Important examples are polynomial rings over the integers or over a field, Euclidean domains and principal ideal domains.
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| 12. | In his paper FAC, Serre asked whether a finitely generated projective module over a polynomial ring is free.
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| 13. | The concept and algorithms of Gr�bner bases have been generalized to submodules of free modules over a polynomial ring.
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| 14. | More generally, the Chevalley Shephard Todd theorem characterizes finite groups whose algebra of invariants is a polynomial ring.
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| 15. | A more general notion of polynomial ring is obtained when the distinction between these two formal products is maintained.
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| 16. | Defined an analogous basis when the polynomial ring is replaced by the ring of exponentials of the weight lattice.
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| 17. | This case is closely related to work of Macaulay on graded polynomial rings and is sometimes called Macaulay duality.
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| 18. | For example, choosing a basis, a symmetric algebra satisfies the universal property and so is a polynomial ring.
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| 19. | Examples of Euclidean domains include fields, polynomial rings in one variable over a field, and the Gaussian integers.
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| 20. | This is equivalent to the statement that the polynomial ring is a free module over the ring of radical polynomials.
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