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Concept
Principal ideal
Summary
In mathematics, specifically ring theory, a principal ideal is an ideal in a ring that is generated by a single element of through multiplication by every element of The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset generated by a single element which is to say the set of all elements less than or equal to in
The remainder of this article addresses the ring-theoretic concept.
a left principal ideal of is a subset of given by for some element
a right principal ideal of is a subset of given by for some element
a two-sided principal ideal of is a subset of given by for some element namely, the set of all finite sums of elements of the form
While this definition for two-sided principal ideal may seem more complicated than the others, it is necessary to ensure that the ideal remains closed under addition.
If is a commutative ring with identity, then the above three notions are all the same.
In that case, it is common to write the ideal generated by as or
Not all ideals are principal.
For example, consider the commutative ring of all polynomials in two variables and with complex coefficients. The ideal generated by and which consists of all the polynomials in that have zero for the constant term, is not principal. To see this, suppose that were a generator for Then and would both be divisible by which is impossible unless is a nonzero constant.
But zero is the only constant in so we have a contradiction.
In the ring the numbers where is even form a non-principal ideal. This ideal forms a regular hexagonal lattice in the complex plane. Consider and These numbers are elements of this ideal with the same norm (two), but because the only units in the ring are and they are not associates.
A ring in which every ideal is principal is called principal, or a principal ideal ring. A principal ideal domain (PID) is an integral domain in which every ideal is principal. Any PID is a unique factorization domain; the normal proof of unique factorization in the integers (the so-called fundamental theorem of arithmetic) holds in any PID.
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