Abstract
This chapter discusses applications of essential extensions, maximal essential extensions, and iterated maximal essential extensions to different topics of radical theory. It presents a theory based on essential covers, intersection property, and special radicals. If M be a regular class of associative rings containing no or all zero rings, The following statements are equivalent. (1) U M is hereditary; (2) Mk ⊆ SU M; (3) UM = UMk; (4) UM has the intersection property relative to Mk; and (5) UM ∩ Mk = 0. A radical α has the intersection property relative to a class M of prime subdirectly irreducible rings if M is a special class and α = UM.
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