Free Ideal Ring - Properties and Examples

Properties and Examples

It turns out that a left and right fir is a domain. Furthermore, a commutative fir is precisely a principal ideal domain, while a commutative semifir is precisely a Bézout domain. These last facts are not generally true for noncommutative rings, however (Cohn 1971).

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 semihereditary ring. Because projective modules over local rings are free, and because local rings have invariant basis number, it follows that a local, right hereditary ring is a right fir, and a local, right semihereditary ring is a right semifir.

Unlike a principal right idea domain, a right fir is not necessarily right Noetherian, however in the commutative case, R is a Dedekind domain since it is a hereditary domain, and so is necessarily Noetherian.

Another important and motivating example of a free ideal ring are the free associative (unital) k-algebras for division rings k, also called non-commutative polynomial rings (Cohn 2000, §5.4).

Semifirs have invariant basis number and also they admit a field of fractions. Every semifir is a Sylvester domain.

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