Unitary Strongly Prime Rings and Related Radicals

A unitary strongly prime ring is defined as a prime ring whose central closure is simple with identity element. The class of unitary strongly prime rings is a special class of rings and the corresponding radical is called the unitary strongly prime radical. In this paper we prove some results on unitary strongly prime rings. The results are applied to study the unitary strongly prime radical of a polynomial ring and also R-disjoint maximal ideals of polynomial rings over R in a finite number of indeterminates. From this we get relations between the Brown-McCoy radical and the unitary strongly prime radical of polynomial rings. In particular, the Brown-McCoy radical of R[X] is equal to the unitary strongly prime radical of R[X] and also equal to S(R)[X], where S(R) denotes the unitary strongly prime radical of R, when X is an infinite set of either commuting or non-commuting indeterminates. For a PI ring R this holds for any set X.