Classical ideals theory of maximal subrings in non-commutative rings
Alborz Azarang
TL;DR
This work analyzes how central ring-theoretic ideals behave under maximal subring extensions in noncommutative rings. It develops a conductor-ideals framework and bracketed decompositions, establishing key equalities such as $Nil_*(R)=Nil_*(T)\cap R$ and $Nil^*(R)=Nil^*(T)\cap R$, and, in the reduced case, a dichotomy $Z({}_RT)=0$ or $T=R\oplus Z({}_RT)$ with $(R:T)=(R:T)_l=(R:T)_r=ann_R(Z({}_RT))$. In the case $T=R\oplus I$, the paper tightly relates radicals, socle, and singular ideals of $R$ and $T$ via contractions and decompositions, and analyzes the prime/semiprime/primitive structure of $T$ in terms of $I$ and its square. For left Artinian maximal subrings, it gives a detailed spectrum-theoretic and module-theoretic picture, including a matrix-ring decomposition $T/(R:T)\cong \mathbb{M}_n(S)$ with $S$ restricted to at most two nonzero proper ideals, yielding a clear dichotomy between $T$ remaining left Artinian or $S$ having a highly constrained ideal structure. Overall, the results provide precise interdependencies among radicals, socles, and centers across maximal subring extensions, with sharp structural classifications in reduced and Artinian settings.
Abstract
Let $R$ be a maximal subring of a ring $T$. In this paper we study relation between some important ideals in the ring extension $R\subseteq T$. In fact, we would like to find some relation between $Nil_*(R)$ and $Nil_*(T)$, $Nil^*(R)$ and $Nil^*(T)$, $J(R)$ and $J(T)$, $Soc({}_RR)$ and $Soc({}_RT)$, and finally $Z({}_RR)$ and $Z({}_RT)$; especially, in certain cases, for example when $T$ is a reduced ring, $R$ (or $T$) is a left Artinian ring, or $R$ is a certain maximal subring of $T$. We show that either $Soc({}_RR)=Soc({}_RT)$ or $(R:T)_r$ (the greatest right ideal of $T$ which is contained in $R$) is a left primitive ideal of $R$. We prove that if $T$ is a reduced ring, then either $Z({}_RT)=0$ or $Z({}_RT)$ is a minimal ideal of $T$, $T=R\oplus Z({}_RT)$, and $(R:T)=(R:T)_l=(R:T)_r=ann_R(Z({}_RT))$. If $T=R\oplus I$, where $I$ is an ideal of $T$, then we completely determine relation between Jacobson radicals, lower nilradicals, upper nilradicals, socle and singular ideals of $R$ and $T$. Finally, we study the relation between previous ideals of $R$ and $T$ when either $R$ or $T$ is a left Artinian ring.