On the Restricted Minimum Condition for Rings
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On the Restricted Minimum Condition for Rings A. Karami Z. and M. R. Vedadi Abstract. Generalizing Artinian rings, a ring R is said to have right restricted minimum condition (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 r.RMC quasi-Frobenius? (ii) Whether a serial ring with r.RMC must be Noetherian? We carry out a study of rings with r.RMC and determine when a right extending ring has r.RMC in terms of rings S M such that S is right Artinian, MQ is semisimple (Q = Q(R)) 0 R and R is a semiprime ring with Krull dimension 1. We proved that a left self-injective ring R with r.RMC is quasi-Frobenius if and only if Zr (R) = Zl (R) if and only if Zr (R) is a finitely generated left ideal and N(R) ∩ Soc(RR ) is a finitely generated right ideal. Right serial rings with r.RMC are studied and proved that a non-singular serial ring has r.RMC if and only if it is a left Noetherian ring. Examples are presented to describe our results and to show that RMC is not symmetric for a ring. Mathematics Subject Classification. Primary: 16P40, 16P20; Secondary: 16D50, 13E10. Keywords. Artinian module, extending module, Noetherian ring, quasi-Frobenius ring, restricted minimum condition, serial ring.
1. Introduction Throughout this paper, rings will have a non-zero identity and modules will be right and unitary. A widely used result of Hopkins and Levitzki states that “every right Artinian ring is a right Noetherian ring”, [9, Theorem 4.15]. This is well known that a commutative Noetherian ring with zero dimension is Artinian. Motivated by these Theorems, Cohen studied and characterized commutative rings R satisfy the restricted minimum condition, called RM rings (i.e., R/I is an Artinian ring for every 0 = I R), see [5, Corollary 1]. 0123456789().: V,-vol
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A. Karami Z. and M. R. Vedadi
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It was proved that a commutative ring R is RM if and only if R is Noetherian and every non-zero prime ideal of R is maximal. In [19], Ornstein continued the study of RM rings (not necessary commutative). Later on, Camillo and Krause asked whether a ring R is right Noetherian if every proper cyclic right R-module is Artinian, [2]. It is known that any ring with latter property is either right Artinian or right Ore domain. Webber in [24, Theorem 4] showed that semiprime Noetherian hereditary rings are RM. Results in [24] were extended by Chatters [3]. Definition 1.1. Following [2,3], an R-module M is said to have the restricted minimum condition (RMC, for short) if for every essential submodule N of M , the factor module M/N is Artinian. Also, a ring R is said to have right restricted minimum condition (r.RMC) if the right R-module R has RM C. Rings with left restricted minimum condition (l.RMC) is defined similarly. Clearly, if M/Soc(M ) is Artinian, then M has RMC. In [3, Theorem 2.2], it is proved that if
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