# Free ideal ring

fir.

A (non-commutative) ring (with unit element) in which all (one-sided) ideals are free. More precisely, a right fir is a ring $R$ in which all right ideals are free of unique rank, as right $R$-modules. A left fir is defined correspondingly. Firs may be regarded as generalizing the notion of a principal ideal domain.

Consider dependence relations of the form $x \cdot y = x _ {1} y _ {1} + {} \dots + x _ {n} y _ {n} = 0$, $x _ {i} , y _ {i} \in R$ ($x$ a row vector, $y$ a column vector). Such a relation is called trivial if for each $i = 1 \dots n$ either $x _ {i} = 0$ or $y _ {i} = 0$. An $n$-term relation $x \cdot y = 0$ is trivialized by an invertible $n \times n$ matrix $M$ if the relation $( xM) ( M ^ {-1} y)$ is trivial. Now let $R$ be a non-zero ring with unit element, then the following properties are all equivalent: i) every $m$-term relation $\sum x _ {i} y _ {i} = 0$, $m \leq n$, can be trivialized by an invertible $m \times m$ matrix; ii) given $x _ {1} \dots x _ {n} \in R$, $m \leq n$, which are right linearly dependent, there exist $( m \times m )$-matrices $M , N$ such that $MN = I _ {m}$ and $( x _ {1} \dots x _ {m} ) M$ has at least one zero component; iii) any right ideal of $R$ generated by $m \leq n$ right linearly dependent elements has fewer than $m$ generators; and iv) any right ideal of $R$ on at most $n$ generators is free of unique rank. These properties are also equivalent to their left-right analogues. There are several more equivalent conditions, cf. [a1].

A ring which satisfies these conditions is called an $n$-fir. A ring which is an $n$-fir for all $n$ is called a semi-fir.

An integral domain $R$ satisfying $aR \cap bR \neq \{ 0 \}$ for all $a , b \in R ^ {*} = R \setminus \{ 0 \}$ (the Ore condition) is called a right Ore ring (cf. also Associative rings and algebras for Ore's theorem). It follows that a ring $R$ is a Bezout domain (cf. Bezout ring) if and only if it is a $2$-fir and a right Ore ring.

For any ring $R$ the following are equivalent: 1) $R$ is a total matrix ring over a semi-fir; 2) $R$ is Morita equivalent (cf. Morita equivalence) to a semi-fir; 3) $R$ is right semi-hereditary (i.e. all finitely-generated right ideals are projective) and $R$ is projective-trivial; and 4) the left-right analogue of 3). Here a ring is projective-trivial if there exists a projective right module $P$ (called the minimal projective of $R$) such that every finitely-projective right module $M$ is the direct sum of $n$ copies of $P$ for some $n$ unique determined by $M$.

For any ring $R$ the following are equivalent: a) $R$ is a total matrix ring over a right fir; b) $R$ is Morita equivalent to a right fir; and c) $R$ is right hereditary (i.e. all right ideals are projective) and projective-trivial.

If $R$ is a semi-fir, then a right module $P$ is flat if and only if every finitely-generated submodule of $P$ is free (i.e. if and only if $P$ is locally free).

How to Cite This Entry:
Free ideal ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Free_ideal_ring&oldid=52050