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In principal ideal domains a near converse holds: every nonzero prime ideal is maximal. All principal ideal domains are integrally closed. The previous three statements give the definition of a Dedekind domain, and hence every principal ideal domain is a Dedekind domain. Let A be an integral domain, the following are equivalent. A is a PID.
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 .
A right ideal is defined similarly, with the condition replaced by . A two-sided ideal is a left ideal that is also a right ideal. If the ring is commutative, the three definitions are the same, and one talks simply of an ideal. In the non-commutative case, "ideal" is often used instead of "two-sided ideal".
Another important example of a DVR is the ring of formal power series = [[]] in one variable over some field .The "unique" irreducible element is , the maximal ideal of is the principal ideal generated by , and the valuation assigns to each power series the index (i.e. degree) of the first non-zero coefficient.
In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules over a principal ideal domain (PID) can be uniquely decomposed in much the same way that integers have a prime factorization.
Every semisimple ring R which is not just a product of fields is a noncommutative right and left principal ideal ring (it need not be a domain, as the example of n x n matrices over a field shows). Every right and left ideal is a direct summand of R , and so is of the form eR or Re where e is an idempotent of R .
An integral domain where every finitely generated ideal is principal (that is, a Bézout domain) satisfies (ACCP) if and only if it is a principal ideal domain. [ 4 ] The ring Z + X Q [ X ] of all rational polynomials with integral constant term is an example of an integral domain (actually a GCD domain) that does not satisfy (ACCP), for the ...
Every principal right ideal domain R is a right fir, since every nonzero principal right ideal of a domain is isomorphic to R. In the same way, a right Bézout domain is a semifir. Since all right ideals of a right fir are free, they are projective. So, any right fir is a right hereditary ring, and likewise a right semifir is a right ...