Star, sharp, core and dual core partial order in rings with involution.

The star, sharp, core and dual core partial orders are already known and investigated in the set of complex matrices. In this paper these orders are extensively studied in the context of arbitrary ring with involution. It is shown that the condition a < b , where < is one of the considered partial orders, defines appropriate idempotents and thus the suitable matrix representations of a and b are obtained. Using these representations we proved that a < b if and only if G ( b ) ⊆ G ( a ) , where G ( a ) is appropriate subset of the set of all generalized inverses of a . If < - is the minus partial order, the conditions under which a < - b implies a < b are determined. Also, several well-known results are generalized and some new results are proved. [ABSTRACT FROM AUTHOR]