Let R be a ring and a, b, c ∈ R. We give a novel characterization of group inverses (resp. EP elements) by the properties of right (resp. left ) c -regular inverses of a and discuss the relation among the strongly left (b, c)-invertibility of a, the right ca-regularity of b , and the (b, c)-invertibility of a. | Turk J Math (2018) 42: 3078 – 3089 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Some characterizations of right c-regularity and (b, c)-inverse Ruju ZHAO∗,, Hua YAO,, Long WANG, Junchao WEI, School of Mathematical Sciences, Yangzhou University, Yangzhou, Jiangsu, . China Received: • Accepted/Published Online: • Final Version: Abstract: Let R be a ring and a, b, c ∈ R . We give a novel characterization of group inverses (resp. EP elements) by the properties of right (resp. left ) c -regular inverses of a and discuss the relation among the strongly left (b, c) -invertibility of a , the right ca -regularity of b , and the (b, c) -invertibility of a . Finally, we investigate the sufficient and necessary condition for a ring to be a strongly left min-Abel ring by means of the (b, c) -inverse of a . Key words: Right c -regular element, (b, c) -inverse, group inverse, EP element, left min-Abel ring 1. Introduction Let S be a semigroup and a, b, c ∈ S . Then a is said to be (b, c) -invertible [4] if there exists y ∈ bSy ∩ ySc such that yab = b and cay = c. Such an y is called a (b, c) -inverse of a, which is always unique if it exists, denoted by a||(b,c) . In [5], Drazin considered the following problem: in any semigroup S (or any associative ring ) with unit element 1, and for any given a ∈ S , the properties 1 ∈ Sa (1 ∈ aS ) of left (right) invertibility are often useful as weaker versions of ordinary two-sided invertibility, and it is natural to seek corresponding one-sided versions for at least some types of generalized invertibility. Hence, Drazin in [5] introduced the left (b, c)-inverse as follows: let S be any semigroup and let a, b, c ∈ S . Then a is said to be left (b, c) -invertible if b ∈ Scab , or equivalently if there exists x ∈ Sc such that xab = b, in which case any such x will be called a left (b, c) -inverse of a. The left (b, c) .
