In this paper, we introduce a new notion which lies properly between strong [Formula: see text]-regularity and pseudopolarity. A ring [Formula: see text] is feckly polar if for any [Formula: see text] there exists [Formula: see text] such that [Formula: see text] Many structure theorems are proved. Further, we investigate feck polarity for triangular matrix and matrix rings. The relations among strongly [Formula: see text]-regular rings, pseudopolar rings and feckly polar rings are also obtained.

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We present new additive results for the group inverse in a Banach algebra under certain perturbations. The upper bound of (a + b) # − a d is thereby given. These extend the main results in [X. Liu, Y. Qin and H. Wei, Perturbation bound of the group inverse and the generalized Schur complement in Banach algebra,