In algebra

Algebra

Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

, a semifir is a ring in which every finitely generated right ideal is free of unique rank. Similarly, a fir is a ring in which every right ideal is free of unique rank. Every fir is thus a semifir. A commutative semifir is a Bézout domain

Bézout domain

In mathematics, a Bézout domain is an integral domain in which the sum of two principal ideals is again a principal ideal. This means that for every pair of elements a Bézout identity holds, and that every finitely generated ideal is principal...

, while a commutative fir is a principal ideal domain

Principal ideal domain

In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...

. The importance of semifir comes from the fact that it (i) has the invariant basis number

Invariant basis number

In mathematics, the invariant basis number property of a ring R is the property that all free modules over R are similarly well-behaved as vector spaces, with respect to the uniqueness of their ranks.-Definition:...

 and (ii) admits the universal field of fractions. Every semifir is a Sylvester domain

Sylvester domain

In mathematics, a Sylvester domain, named after James Joseph Sylvester by , is a ring in which Sylvester's law of nullity holds. This means that if A is an m by n matrix and B an n by s matrix over R, thenwhere ρ is the inner rank of a matrix...

.

The notion of semifirs was introduced by P. M. Cohn.