The conductor ideals of maximal subrings in non-commutative rings

Articles in Press

Document Type : Research Paper

Author

Department of Mathematics, Faculty of Mathematical Sciences and Computer, Shahid Chamran University of Ahvaz, Ahvaz-Iran

10.48308/cgasa.2026.240488.1556

Abstract

Let $R$ be a maximal subring of a ring $T$, and $(R:T)$, $(R:T)_\ell$ and $(R:T)_r$ denote the largest ideal, left ideal and right ideal of $T$ that are contained in $R$, respectively. It is shown that both $(R:T)_\ell$ and $(R:T)_r$ are prime ideals of $R$, and $|{\rm Min}_R((R:T))|\leq 2$. We prove that if $T_R$ has a maximal submodule, then $(R:T)_\ell$ is a right primitive ideal of $R$. We investigate the conditions under which $(R:T)_r$ is a completely prime (right) ideal of $R$ or of $T$. We show that Char$(R/(R:T)_\ell)={\rm Char}(R/(R:T)_r)$, and if Char$(T)$ is neither zero nor a prime number, then $(R:T)\neq 0$. When $|{\rm Min}(R)|\geq 3$, both $(R:T)$ and $(R:T)_\ell(R:T)_r$ are nonzero ideals. Assuming $R$ is integrally closed in $T$, we prove that $(R:T)_\ell$ and $(R:T)_r$ are prime one-sided ideals of $T$; moreover $(R:T)$ is a semiprime ideal of $T$ and either $(R:T)$ is a prime ideal of $T$ or $(R:T)=(R:T)_\ell\cap (R:T)_r$ is a semiprime ideal of $R$. We observe that if $(R:T)_lT=T$, then $T$ is a finitely generated left $R$-module and $(R:T)_\ell$ is a finitely generated right $R$-module which is also a right primitive ideal of $R$. Finally, we study the transfer of the Noetherian and the Artinian properties between $R$ and $T$.

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References

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Categories and General Algebraic Structures with Applications

Articles in Press, Accepted Manuscript
Available Online from 24 April 2026

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