Generalizing Artinian rings, a ring R is said to have right restricted minimum condition (\({\mathrm{r.RMC}}\), for short) if R/A is an Artinian right R-module for any essential right ideal A of R. It is asked in Jain et al. [Cyclic Modules and the Structure of Rings, Oxford University Press, Oxford, 2012, 3.17 Questions (2)] that (i) Is a left self-injective ring with \({\mathrm{r.RMC}}\) quasi-Frobenius? (ii) Whether a serial ring with \({\mathrm{r.RMC}}\) must be Noetherian? We carry out a study of rings with \({\mathrm{r.RMC}}\) and determine when a right extending ring has \({\mathrm{r.RMC}}\) in terms of rings \({\begin{bmatrix} S&{}\quad M\\ 0&{}\quad R\end{bmatrix}}\) such that S is right Artinian, \(M_{Q}\) is semisimple (\(Q={\mathrm{Q}}(R)\)) and R is a semiprime ring with Krull dimension 1. We proved that a left self-injective ring R with \({\mathrm{r.RMC}}\) is quasi-Frobenius if and only if \(\hbox {Z}_{r}(R) = \hbox {Z}_{l}(R)\) if and only if \(\hbox {Z}_{r}(R)\) is a finitely generated left ideal and \({\mathrm{N}}(R)\cap {\mathrm{Soc}}(R_{R})\) is a finitely generated right ideal. Right serial rings with \({\mathrm{r.RMC}}\) are studied and proved that a non-singular serial ring has \({\mathrm{r.RMC}}\) if and only if it is a left Noetherian ring. Examples are presented to describe our results and to show that \(\mathrm{RMC}\) is not symmetric for a ring.

泛化Artinian环，如果R / A是任何基本右心理想A的Artinian右心R-模，则说成环R具有权限制的最小条件（简称（\（{\ mathrm {r.RMC}} \\））。的[R 。在Jain等人中有问。[环模和环的结构，牛津大学出版社，牛津，2012，3.17问题（2）]：（i）是左自注入环，具有\（{\ mathrm {r.RMC}} \）准- Frobenius？（ii）是否带有\（{\ mathrm {r.RMC}} \）的串行环必须是Noetherian？我们使用\（{\ mathrm {r.RMC}} \）对环进行研究，并确定何时向右延伸的环具有\（{\ mathrm {r.RMC}} \）在环的术语\（{\开始{bmatrix} S＆{} \四中号\\ 0＆{} \四r \ {端bmatrix}} \），使得小号是正确的Artinian，\（M_ {Q} \）是半简单的（\（Q = {\ mathrm {Q}}（R）\）），R是Krull尺寸为1的半素环。我们证明了左自当且仅当\（\ hbox {Z} _ {r}（R）= \ hbox {Z} _ {l}（）时，带有\（{\ mathrm {r.RMC}} \）的内射环R是准弗罗贝纽斯R）\）仅当\（\ hbox {Z} _ {r}（R）\）是有限生成的左理想值和\（{\ mathrm {N}}（R）\ cap {\ mathrm {Soc }}（R_ {R}）\）是有限生成的右理想。右串行环与研究了\（{\ mathrm {r.RMC}} \\，并且证明非奇异串行环具有且仅当是左Noetherian环时才具有\（{\ mathrm {r.RMC}} \\）。给出了一些例子来描述我们的结果，并表明\（\ mathrm {RMC} \）对于一个环而言不是对称的。