THE FINITE DIMENSIONAL PRIME RINGS

If R is ring and M is right (or left) R-module, then M is called faithful R-module if, for some in R, x.a=0 for all x.mem.M then a=0. In [4], R.E. Johnson defines that M is prime module if every non-zero submodule of M is faithful. Let us define that M is of prime type provided that M is faithful if and only if every non-zero submodule is faithful. We call right (left) ideal I of R is of prime type if R/I is of prime type as R-module. This is equivalent to the condition that if xRy.subeq.I then either x.mem.I ro y.mem.I (see [5:3:1]). It is easy to see that in case R is commutative ring then right or left ideal of prime type is just prime ideal. We have defined in [5], that chain of right ideals of prime type in ring R is finite strictly increasing sequence I.contnd.I.contnd....contnd.I; the length of the chain is n. By the right dimension of ring R, which is denoted by dim, R, we mean the supremum of the length of all chains of right ideals of prime type in R. It is an integer .geq.0 or .inf.. The left dimension of R, which is denoted by dim R is similarly defined. It was shown in [5], that dimR=0 if and only if dim R=0 if and only if R modulo the prime radical is strongly regular ring. By a strongly regular ring, we mean that for every in R there is x in R such that axa=a=ax. It was also shown that R is simple ring if and only if every right ideal is of prime type if and only if every left ideal is of prime type. In case, R is (right or left) primitive ring then dimR=n if and only if dim R=n if and only if R.iden.D , n+1 by n+1 matrix ring on division ring D. in this paper, we establish the following results: (1) If R is prime ring and dimR=n then either R is righe Ore domain such that every non-zero right ideal of prime type contains non-zero minimal prime ideal or the classical ring of ritght quotients is isomorphic to m*m matrix ring over division ring where m.leq.n+1. (b) If R is prime ring and dimR=n then dim R=n if dim R=n if dim RR=1 then R is an unique factorization domain.TEX>R=1 then R is an unique factorization domain.