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The set of all polynomials with real coefficients that are divisible by the polynomial + is an ideal in the ring of all real-coefficient polynomials [] . Take a ring R {\displaystyle R} and positive integer n {\displaystyle n} .
The ring of power series can also be seen as the ring completion of the polynomial ring with respect to the ideal ... The skew-polynomial ring is defined ...
An important ideal of the ring called the Jacobson radical can be defined using maximal right (or maximal left) ideals. If R is a unital commutative ring with an ideal m, then k = R/m is a field if and only if m is a maximal ideal. In that case, R/m is known as the residue field. This fact can fail in non-unital rings.
In the ring [] of all polynomials with integer coefficients, the ideal generated by 2 and X is a prime ideal. The ideal consists of all polynomials constructed by taking 2 times an element of Z [ X ] {\displaystyle \mathbb {Z} [X]} and adding it to X times another polynomial in Z [ X ] {\displaystyle \mathbb {Z} [X]} (which converts the ...
Let be a field and = [] be the polynomial ring over with n indeterminates =,, …,.. A monomial in is a product = for an n-tuple = (,, …,) of nonnegative integers.. The following three conditions are equivalent for an ideal :
The ideal quotient is useful for calculating primary decompositions. It also arises in the description of the set difference in algebraic geometry (see below). (I : J) is sometimes referred to as a colon ideal because of the notation. In the context of fractional ideals, there is a related notion of the inverse of a fractional ideal.
[]: the ring of all polynomials with integer coefficients. It is not principal because 2 , x {\displaystyle \langle 2,x\rangle } is an ideal that cannot be generated by a single polynomial. K [ x , y , … ] , {\displaystyle K[x,y,\ldots ],} the ring of polynomials in at least two variables over a ring K is not principal, since the ideal x , y ...
In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many primary ideals (which are related to, but not quite the same as, powers of prime ideals).