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glebovg
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Can anyone explain ideals and polynomial rings i.e. definitions, examples, the most important theorems, etc.?
glebovg said:Can anyone explain ideals and polynomial rings i.e. definitions, examples, the most important theorems, etc.
alexfloo said:A basis of an ideal is a set of polynomials that "generate" the ideal; that is, if f1, .., fs is a basis for I, then each f in I can be written as h1f1 + h2f2 + ... + hsfs. A basis "spans" an ideal in a similar sense to how a linear basis spans a vector space.
Every ideal of a polynomial ring has a finite basis; this is called the Hilbert Basis Theorem.
Any ideal of polynomials in one variable can be generated by a single element. The ideal generated by f, a polynomial in x, is the set of polynomials which are divisible by f. Therefore, we can check for membership in the ideal just be computing that single polynomial generator, and then dividing (polynomial long division, which you may recall from high school, and writes g = qf+r, to divide g by our generator f). The remainder r is zero if and only if g is in the ideal generated by f.
In larger numbers of variables, this fails. There may be more than one generator; this means there are several different ways we can write "f = h1f1 + h2f2 + ... + hsfs + r," with several different remainders r. (There are algorithms for "dividing by multiple polynomials at the same time," or "dividing by a basis.") If we find one of these remainders, and it is zero, then we definitely have membership; if we find one and it is not zero, however, it doesn't mean that there is not a different way where the remainder is zero. We might hope for an alternative.
Groebner bases are a special basis, which every polynomial ideal has, with the extremely useful property that dividing by a Groebner basis gives remainder zero if and only if the dividend is in the ideal. This makes them very useful objects in commutative algebra and algebraic geometry.
alexfloo said:A basis of an ideal is a set of polynomials that "generate" the ideal; that is, if f1, .., fs is a basis for I, then each f in I can be written as h1f1 + h2f2 + ... + hsfs. A basis "spans" an ideal in a similar sense to how a linear basis spans a vector space.
*** Not so "similar": in an ideal we can multiply by elements of the ring AND ALSO by other elements of the ideal itself. ***
Every ideal of a polynomial ring has a finite basis; this is called the Hilbert Basis Theorem.
*** Some care's needed here: this is true if the polynomial ring is over a Noetherian ring (a field, say), NOT in general. ***
Any ideal of polynomials in one variable can be generated by a single element.
*** Again, more care is needed when writing down these results: this is far from being true in general, but it
is true if, for example, the polynomial ring is over a field, say.
DonAntonio ***
The ideal generated by f, a polynomial in x, is the set of polynomials which are divisible by f. Therefore, we can check for membership in the ideal just be computing that single polynomial generator, and then dividing (polynomial long division, which you may recall from high school, and writes g = qf+r, to divide g by our generator f). The remainder r is zero if and only if g is in the ideal generated by f.
In larger numbers of variables, this fails. There may be more than one generator; this means there are several different ways we can write "f = h1f1 + h2f2 + ... + hsfs + r," with several different remainders r. (There are algorithms for "dividing by multiple polynomials at the same time," or "dividing by a basis.") If we find one of these remainders, and it is zero, then we definitely have membership; if we find one and it is not zero, however, it doesn't mean that there is not a different way where the remainder is zero. We might hope for an alternative.
Groebner bases are a special basis, which every polynomial ideal has, with the extremely useful property that dividing by a Groebner basis gives remainder zero if and only if the dividend is in the ideal. This makes them very useful objects in commutative algebra and algebraic geometry.
An ideal in a polynomial ring is a subset of the ring that satisfies certain algebraic properties. Specifically, it is a set of polynomials that can be multiplied by any polynomial in the ring to produce another polynomial in the set. In other words, it is a set of polynomials that is closed under multiplication by elements in the ring.
Ideals are intimately related to polynomial rings, as they are used to define the structure of the ring. In fact, polynomial rings are defined as the quotient of a polynomial ring by an ideal. This means that every polynomial in the ring can be expressed as the sum of a polynomial in the ideal and a polynomial in the ring's ideal complement.
A principal ideal is generated by a single element, while a non-principal ideal is generated by multiple elements. In other words, a principal ideal can be written as the set of all multiples of a single polynomial, while a non-principal ideal requires multiple polynomials to generate it.
Ideals play a central role in algebraic geometry, as they are used to define algebraic varieties. In this context, an ideal is associated with a set of points in n-dimensional space, where n is the number of variables in the polynomial ring. The solutions to the polynomials in the ideal correspond to the points in the variety.
Yes, ideals can be used to solve polynomial equations. In fact, the ideal generated by a polynomial equation represents all possible solutions to that equation. By finding the ideal and its associated variety, one can determine all solutions to the polynomial equation.