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Mathematics > Rings and Algebras
Title: On some counterparts of Rickart $*$-algebras
(Submitted on 17 Oct 2023 (v1), last revised 25 Apr 2024 (this version, v4))
Abstract: 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 $\mathcal{N}$.
Submission history
From: Farhodjon Arzikulov [view email][v1] Tue, 17 Oct 2023 18:35:25 GMT (10kb)
[v2] Fri, 29 Dec 2023 19:29:09 GMT (10kb)
[v3] Tue, 12 Mar 2024 16:39:48 GMT (11kb)
[v4] Thu, 25 Apr 2024 19:03:47 GMT (11kb)
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