On some counterparts of Rickart $*$-algebras
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866929327737667584 |
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| author | Arzikulov, Farhodjon Khakimov, Utkirbek |
| author_facet | Arzikulov, Farhodjon Khakimov, Utkirbek |
| contents | In the present paper, we introduce and study counterparts of Rickart involutive algebras, i.e., almost inner Rickart algebras. We prove that a nilpotent associative algebra, which has no nilpotent elements with nonzero square roots, is an almost inner Rickart algebra. A nilpotent associative algebra, which has no nilpotent elements with a square root $b$ such that $b^3\neq 0$, is not an almost inner Rickart algebra if there exists a nonzero element $a$ such that $a^2\neq 0$. As a main result of the paper, we describe a finite-dimensional almost inner Rickart algebra $\mathcal{A}$ over a field $\mathbb{F}$, isomorphic to $\mathbb{F}^n\dot{+} \mathcal{N}$, $n=1,2$, with a nilradical $\mathcal{N}$. Also, we classify finite-dimensional almost inner Rickart algebras over the real or complex numbers with a nonzero nilradical $\mathcal{N}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_11519 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On some counterparts of Rickart $*$-algebras Arzikulov, Farhodjon Khakimov, Utkirbek Rings and Algebras Operator Algebras 16W10, 16E50, 16N40 In the present paper, we introduce and study counterparts of Rickart involutive algebras, i.e., almost inner Rickart algebras. We prove that a nilpotent associative algebra, which has no nilpotent elements with nonzero square roots, is an almost inner Rickart algebra. A nilpotent associative algebra, which has no nilpotent elements with a square root $b$ such that $b^3\neq 0$, is not an almost inner Rickart algebra if there exists a nonzero element $a$ such that $a^2\neq 0$. As a main result of the paper, we describe a finite-dimensional almost inner Rickart algebra $\mathcal{A}$ over a field $\mathbb{F}$, isomorphic to $\mathbb{F}^n\dot{+} \mathcal{N}$, $n=1,2$, with a nilradical $\mathcal{N}$. Also, we classify finite-dimensional almost inner Rickart algebras over the real or complex numbers with a nonzero nilradical $\mathcal{N}$. |
| title | On some counterparts of Rickart $*$-algebras |
| topic | Rings and Algebras Operator Algebras 16W10, 16E50, 16N40 |
| url | https://arxiv.org/abs/2310.11519 |