### Abstract

We characterize the elements with outer inverse in a semigroup S, and provide explicit expressions for the class of outer inverses b of an element a such that bS⊆yS and Sb⊆Sx, where x, y are any arbitrary elements of S. We apply this result to characterize pairs of outer inverses of given elements from an associative ring R, satisfying absorption laws extended for the outer inverses. We extend the result on right–left symmetry of aR⊕bR=(a+b)R (Jain–Prasad, 1998) to the general case of an associative ring. We conjecture that ‘given an outer inverse x of a regular element a in a semigroup S, there exists a reflexive generalized inverse y of a such that x≤^{−}y' and prove the conjecture when S is an associative ring.

Original language | English |
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Pages (from-to) | 171-184 |

Number of pages | 14 |

Journal | Linear Algebra and Its Applications |

Volume | 528 |

DOIs | |

Publication status | Published - 01-09-2017 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics

### Cite this

*Linear Algebra and Its Applications*,

*528*, 171-184. https://doi.org/10.1016/j.laa.2016.06.045

}

*Linear Algebra and Its Applications*, vol. 528, pp. 171-184. https://doi.org/10.1016/j.laa.2016.06.045

**Outer inverses : Characterization and applications.** / Bapat, Ravindra B.; Jain, Surender Kumar; Karantha, K. Manjunatha Prasad; Raj, M. David.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Outer inverses

T2 - Characterization and applications

AU - Bapat, Ravindra B.

AU - Jain, Surender Kumar

AU - Karantha, K. Manjunatha Prasad

AU - Raj, M. David

PY - 2017/9/1

Y1 - 2017/9/1

N2 - We characterize the elements with outer inverse in a semigroup S, and provide explicit expressions for the class of outer inverses b of an element a such that bS⊆yS and Sb⊆Sx, where x, y are any arbitrary elements of S. We apply this result to characterize pairs of outer inverses of given elements from an associative ring R, satisfying absorption laws extended for the outer inverses. We extend the result on right–left symmetry of aR⊕bR=(a+b)R (Jain–Prasad, 1998) to the general case of an associative ring. We conjecture that ‘given an outer inverse x of a regular element a in a semigroup S, there exists a reflexive generalized inverse y of a such that x≤−y' and prove the conjecture when S is an associative ring.

AB - We characterize the elements with outer inverse in a semigroup S, and provide explicit expressions for the class of outer inverses b of an element a such that bS⊆yS and Sb⊆Sx, where x, y are any arbitrary elements of S. We apply this result to characterize pairs of outer inverses of given elements from an associative ring R, satisfying absorption laws extended for the outer inverses. We extend the result on right–left symmetry of aR⊕bR=(a+b)R (Jain–Prasad, 1998) to the general case of an associative ring. We conjecture that ‘given an outer inverse x of a regular element a in a semigroup S, there exists a reflexive generalized inverse y of a such that x≤−y' and prove the conjecture when S is an associative ring.

UR - http://www.scopus.com/inward/record.url?scp=84996743422&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84996743422&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2016.06.045

DO - 10.1016/j.laa.2016.06.045

M3 - Article

AN - SCOPUS:84996743422

VL - 528

SP - 171

EP - 184

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

ER -