ON THE FINITE GOLDIE DIMENSION OF SUM OF TWO IDEALS OF AN R-GROUP

Tapatee Sahoo; Babushri Srinivas Kedukodi; Panackal Harikrishnan; Syam Prasad Kuncham

doi:10.7151/dmgaa.1419

ON THE FINITE GOLDIE DIMENSION OF SUM OF TWO IDEALS OF AN R-GROUP

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Abstract

We consider an R-group G, where R is a zero symmetric right nearring. We obtain the Ω-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 Ω of G with Ω-finite Goldie dimension (ΩF GD), if K₁, K₂ are ideals of G wherein K₁ ∩ K₂ is an Ω-complement, then dim_Ω(K₁ + K₂) = dim_Ω(K₁) + dim_Ω(K₂) − dim_Ω(K₁ ∩ K₂). In the sequel, we prove several properties.

Original language	English
Pages (from-to)	177-187
Number of pages	11
Journal	Discussiones Mathematicae - General Algebra and Applications
Volume	43
Issue number	2
DOIs	https://doi.org/10.7151/dmgaa.1419
Publication status	Published - 2023

All Science Journal Classification (ASJC) codes

Algebra and Number Theory
Applied Mathematics

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title = "ON THE FINITE GOLDIE DIMENSION OF SUM OF TWO IDEALS OF AN R-GROUP",

abstract = "We consider an R-group G, where R is a zero symmetric right nearring. We obtain the Ω-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 Ω of G with Ω-finite Goldie dimension (ΩF GD), if K1, K2 are ideals of G wherein K1 ∩ K2 is an Ω-complement, then dimΩ(K1 + K2) = dimΩ(K1) + dimΩ(K2) − dimΩ(K1 ∩ K2). In the sequel, we prove several properties.",

author = "Tapatee Sahoo and Kedukodi, \{Babushri Srinivas\} and Panackal Harikrishnan and Kuncham, \{Syam Prasad\}",

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T1 - ON THE FINITE GOLDIE DIMENSION OF SUM OF TWO IDEALS OF AN R-GROUP

AU - Sahoo, Tapatee

AU - Kedukodi, Babushri Srinivas

AU - Harikrishnan, Panackal

AU - Kuncham, Syam Prasad

PY - 2023

Y1 - 2023

N2 - We consider an R-group G, where R is a zero symmetric right nearring. We obtain the Ω-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 Ω of G with Ω-finite Goldie dimension (ΩF GD), if K1, K2 are ideals of G wherein K1 ∩ K2 is an Ω-complement, then dimΩ(K1 + K2) = dimΩ(K1) + dimΩ(K2) − dimΩ(K1 ∩ K2). In the sequel, we prove several properties.

AB - We consider an R-group G, where R is a zero symmetric right nearring. We obtain the Ω-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 Ω of G with Ω-finite Goldie dimension (ΩF GD), if K1, K2 are ideals of G wherein K1 ∩ K2 is an Ω-complement, then dimΩ(K1 + K2) = dimΩ(K1) + dimΩ(K2) − dimΩ(K1 ∩ K2). In the sequel, we prove several properties.

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ON THE FINITE GOLDIE DIMENSION OF SUM OF TWO IDEALS OF AN R-GROUP

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