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Title:
On the finite Goldie dimension of sum of two ideals of an R-group
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Discussiones Mathematicae - General Algebra and Applications 43(2) (2023) 177-187
Received: 2021-06-19 , Revised: 2021-11-27 , Accepted: 2021-11-27 , Available online: 2023-01-11 , https://doi.org/10.7151/dmgaa.1419
Abstract:
We consider an $R$-group $G,$ where $R$ is a zero symmetric right nearring.
We obtain the $\Omega$-dimension of sum of two ideals of $G$, as a natural
generalization of sum of two subspaces of a finite dimensional vector space;
indeed, difficulty due to non-linearity in $ G. $ However, in this paper we
overcome the situation under a suitable assumption. More precisely, we prove
that for a proper ideal $\Omega$ of $G$ with $\Omega$-finite Goldie dimension
($\Omega$-$FGD$), if $K_1, K_2$ are ideals of $G$ wherein $K_1\cap K_2$ is an
$\Omega$-complement, then $dim_{\Omega}(K_1+K_2)=dim_{\Omega}(K_1)+
dim_{\Omega}(K_2)-dim_{\Omega}(K_1\cap K_2).$ In the sequel, we prove several
properties.
Keywords:
nearring, essential ideal, uniform ideal, finite dimension
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