Content deleted Content added
No edit summary |
m Reverted edits by 63.188.105.66 to last version by Gauss |
||
Line 1:
In [[abstract algebra]], a '''polynomial ring''' is the [[set]] of [[polynomial]]s in one or more variables with coefficients in a [[commutative ring]].
More precisely, let ''R'' be a commutative ring. The polynomial ring in ''n variables,''X<sub>1</sub>'', ..., ''X<sub>n</sub>'', is the set of all polynomials
in those variables with coefficients in ''R''. This ring is denoted ''R''[''X<sub>1</sub>'', ..., ''X<sub>n</sub>'']. For example, an '''integer polynomial''' is a polynomial with coefficients in the ring ''Z'' of [[integer]]s. This is something different from an [[integer-valued polynomial]]. Every [[commutative ring]] that is a finitely-generated [[algebra over a field]] can be written as a [[quotient ring|quotient]] of a polynomial ring.
The definition of a polynomial ring also works for noncommutative [[ring (mathematics)|rings]]. The variables all commute with each other, and with each element of ''R''. You can also define a ring where the variables do <i>not</i> commute with each other. This is known as the [[free algebra]] over ''R''.
Polynomial rings are studied in the field of [[Commutative algebra]].
== Properties ==
* If ''R'' is a field, then ''R''[''X''] is a [[principal ideal ___domain]] (and even a [[Euclidean ___domain]]).
* If ''R'' is a [[unique factorization ___domain]], so is ''R''[''X<sub>1</sub>'', ..., ''X<sub>n</sub>''].
* If ''R'' is an [[integral ___domain]], so is ''R''[''X<sub>1</sub>'', ..., ''X<sub>n</sub>''].
* If ''R'' is [[Noetherian ring|Noetherian]], then ''R''[''X<sub>1</sub>'', ..., ''X<sub>n</sub>''] is Noetherian. This is the [[Hilbert basis theorem]].
| |||