Generalized inverses, ideals, and projectors in rings
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2023
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866913581319061504 |
|---|---|
| author | Morillas, Patricia Mariela |
| author_facet | Morillas, Patricia Mariela |
| contents | The theory of generalized inverses of matrices and operators is closely connected with projections, i.e., idempotent (bounded) linear transformations. We show that a similar situation occurs in any associative ring $\mathcal{R}$ with a unit $1 \neq 0$. We prove that generalized inverses in $\mathcal{R}$ are related to idempotent group endomorphisms $ρ: \mathcal{R} \rightarrow \mathcal{R}$, called projectors. We use these relations to give characterizations and existence conditions for $\{1\}$, $\{2\}$, and $\{1,2\}$-inverses with any given principal/annihilator ideals. As a consequence, we obtain sufficient conditions for any right/left ideal of $\mathcal{R}$ to be a principal or an annihilator ideal of an idempotent element of $\mathcal{R}$. We also study some particular generalized inverses: Drazin and $(b,c)$ inverses, and $(e,f)$ Moore-Penrose, $e$-core, $f$-dual core, $w$-core, dual $v$-core, right $w$-core, left dual $v$-core, and $(p,q)$ inverses in rings with involution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_06149 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Generalized inverses, ideals, and projectors in rings Morillas, Patricia Mariela Rings and Algebras 16U90, 16W10, 15A09 The theory of generalized inverses of matrices and operators is closely connected with projections, i.e., idempotent (bounded) linear transformations. We show that a similar situation occurs in any associative ring $\mathcal{R}$ with a unit $1 \neq 0$. We prove that generalized inverses in $\mathcal{R}$ are related to idempotent group endomorphisms $ρ: \mathcal{R} \rightarrow \mathcal{R}$, called projectors. We use these relations to give characterizations and existence conditions for $\{1\}$, $\{2\}$, and $\{1,2\}$-inverses with any given principal/annihilator ideals. As a consequence, we obtain sufficient conditions for any right/left ideal of $\mathcal{R}$ to be a principal or an annihilator ideal of an idempotent element of $\mathcal{R}$. We also study some particular generalized inverses: Drazin and $(b,c)$ inverses, and $(e,f)$ Moore-Penrose, $e$-core, $f$-dual core, $w$-core, dual $v$-core, right $w$-core, left dual $v$-core, and $(p,q)$ inverses in rings with involution. |
| title | Generalized inverses, ideals, and projectors in rings |
| topic | Rings and Algebras 16U90, 16W10, 15A09 |
| url | https://arxiv.org/abs/2304.06149 |