Additive properties and absorption laws for generalized inverses
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916886466265088 |
|---|---|
| author | Zhou, Yukun Chen, Jianlong Thome, Nestor |
| author_facet | Zhou, Yukun Chen, Jianlong Thome, Nestor |
| contents | Let $a,~f$ be elements in a ring with pseudo core inverses $a^{\scriptsize\textcircled{\tiny D}}$, $f^{\scriptsize\textcircled{\tiny D}}$, and let $b=f-a$. We prove that the absorption law $a^{\scriptsize\textcircled{\tiny D}}(a+f)f^{\scriptsize\textcircled{\tiny D}}=a^{\scriptsize\textcircled{\tiny D}}+f^{\scriptsize\textcircled{\tiny D}}$ holds if and only if $1+a^{\scriptsize\textcircled{\tiny D}}b$ is invertible and the additive property $f^{\scriptsize\textcircled{\tiny D}}=(1+a^{\scriptsize\textcircled{\tiny D}}b)^{-1}a^{\scriptsize\textcircled{\tiny D}}$ is satisfied. We further characterize these properties and establish analogous results for other generalized inverses. Finally, we apply these results to the case of complex matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_04363 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Additive properties and absorption laws for generalized inverses Zhou, Yukun Chen, Jianlong Thome, Nestor Rings and Algebras Let $a,~f$ be elements in a ring with pseudo core inverses $a^{\scriptsize\textcircled{\tiny D}}$, $f^{\scriptsize\textcircled{\tiny D}}$, and let $b=f-a$. We prove that the absorption law $a^{\scriptsize\textcircled{\tiny D}}(a+f)f^{\scriptsize\textcircled{\tiny D}}=a^{\scriptsize\textcircled{\tiny D}}+f^{\scriptsize\textcircled{\tiny D}}$ holds if and only if $1+a^{\scriptsize\textcircled{\tiny D}}b$ is invertible and the additive property $f^{\scriptsize\textcircled{\tiny D}}=(1+a^{\scriptsize\textcircled{\tiny D}}b)^{-1}a^{\scriptsize\textcircled{\tiny D}}$ is satisfied. We further characterize these properties and establish analogous results for other generalized inverses. Finally, we apply these results to the case of complex matrices. |
| title | Additive properties and absorption laws for generalized inverses |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2508.04363 |