A note on regular elements, inner-inverses and summands
Dinesh Khurana, Shubham Mittal · Journal of Algebra and Its Applications · 2025
Suppose that [Formula: see text] and [Formula: see text] are elements of a, not necessarily unital, ring [Formula: see text] such that [Formula: see text] is regular. We prove that if [Formula: see text], then [Formula: see text] implies [Formula: see text] if and only if [Formula: see text]. This salvages Theorem 17 in [R. B. Bapat, S. K. Jain, K. M. P. Karantha and M. D. Raj, Outer inverses: Characterization and applications, Linear Algebra Appl. 528 (2017) 171–184] and extends Theorem 1 in [S. K. Jain and K. M. Prasad, Right-left symmetry of [Formula: see text] in regular rings, J. Pure Appl. Algebra 133 (1998) 141–142] to non-unital rings. For any positive integer [Formula: see text], if [Formula: see text] for [Formula: see text] and [Formula: see text] is regular, then we prove that [Formula: see text] implies [Formula: see text]. For any positive integer [Formula: see text], if [Formula: see text] for [Formula: see text] are such that [Formula: see text] and [Formula: see text], then we prove that the regularity of any [Formula: see text] elements of the set [Formula: see text] implies the regularity of the remaining element. Let [Formula: see text] and [Formula: see text] be regular elements of a ring [Formula: see text]. We prove that [Formula: see text] if and only if [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the set of all inner inverses of the regular element [Formula: see text] in [Formula: see text]. This extends Theorem 2.2 in [T. K. Lee, Unit-regularity of elements in rings, J. Algebra Appl. 22 (2023) 2350135] where the result is proved assuming [Formula: see text] to be semiprime. We also give some applications of Jain–Prasad symmetry result Theorem 1 in [S. K. Jain and K. M. Prasad, Right-left symmetry of [Formula: see text] in regular rings, J. Pure Appl. Algebra 133 (1998) 141–142] by giving new characterizations of unit-regular and special clean elements. This leads to quick proofs of some known results.