The minimal prime spectrum of rings with annihilator conditions

In this paper, we study rings with the annihilator condition (a.c.) and rings whose space of minimal prime ideals, Min(R), is compact. We begin by extending the definition of (a.c.) to noncommutative rings. We then show that several extensions over semiprime rings have (a.c.). Moreover, we investigate the annihilator condition under the formation of matrix rings and classical quotient rings. Finally, we prove that if RR is a reduced ring then: the classical right quotient ring Q(R)Q(R) is strongly regular if and only if RR has a Property (A) and Min(R) is compact, if and only if RR has (a.c.) and Min(R) is compact. This extends several results about commutative rings with (a.c.) to the noncommutative setting.