On some Results Related to Köthe's Conjecture

In this paper, we obtain a partial solution to the following question of Köthe: For which rings R is it true that every left (or both left and right) R-module is a direct sum of cyclic modules? Let R be a ring in which all idempotents are central. We prove that if R is a left Köthe ring (i.e., every left R-module is a direct sum of cyclic modules), then R is an Artinian principal right ideal ring. Consequently, R is a Köthe ring (i.e., each left and each right R-module is a direct sum of cyclic modules) if and only if R is an Artinian principal ideal ring. This is a generalization of a Köthe-Cohen-Kaplansky theorem.

Proceedings of the American Mathematical Society, 2014

In this paper, we obtain a partial solution to the following question of Köthe: For which rings R R is it true that every left (or both left and right) R R -module is a direct sum of cyclic modules? Let R R be a ring in which all idempotents are central. We prove that if R R is a left Köthe ring (i.e., every left R R -module is a direct sum of cyclic modules), then R R is an Artinian principal right ideal ring. Consequently, R R is a Köthe ring (i.e., each left and each right R R -module is a direct sum of cyclic modules) if and only if R R is an Artinian principal ideal ring. This is a generalization of a Köthe-Cohen-Kaplansky theorem.

Bulletin des Sciences Mathématiques

A celebrated conjecture of Auslander and Reiten claims that a finitely generated module M that has no extensions with M ⊕ Λ over an Artin algebra Λ must be projective. This conjecture is widely open in general, even for modules over commutative Noetherian local rings. Over such rings, we prove that a large class of ideals satisfy the extension condition proposed in the aforementioned conjecture of Auslander and Reiten. Along the way we obtain a new characterization of regularity in terms of the injective dimensions of certain ideals. Auslander and Reiten [4] proved that the Generalized Nakayama Conjecture is true if and only if the following conjecture is true: Conjecture 1.2. If Λ is an Artin algebra, then every finitely generated Λ-module M that is a generator (i.e., Λ is a direct summand of a finite direct sum of copies of M) and satisfies Ext i A (M, M) = 0 for all i ≥ 1 must be projective. Auslander, Ding and Solberg [5] formulated the following conjecture, which is equivalent to Conjecture 1.2 over Noetherian rings. Conjecture 1.3. Let M be a finitely generated left module over a left Noetherian ring R. If Ext i R (M, M ⊕ R) = 0 for all i ≥ 1, then M is projective. The case where the ring in Conjecture 1.3 is an Artin algebra is known as the Auslander-Reiten Conjecture. Conjecture 1.3 is known to hold for several classes of rings, for example for Artin algebras of finite representation type [4], however it is widely open in general, even for commutative Gorenstein local rings; see [12]. The purpose of this paper is to exploit a beautiful result of Burch [10] and prove that a large class of weakly m-full ideals satisfy the vanishing condition proposed in Conjecture 1.3 over commuative Noetherian local rings. The definition of a weakly m-full ideal is given in Definition 3.7. Examples of weakly m-full ideals, in fact those I with I : m = I, are abundant in the literature; see, for example, Examples 3.8 and 3.10. The main consequence of our argument can be stated as follows; see Theorem 2.17 and Corollary 3.14. Theorem 1.4. Let (R, m) be a commutative Noetherian local ring and let I be a weakly m-full ideal of R such that I : m = I (or equivalently depth(R/I) = 0.) If R is not regular, then Ext n R (I, I) and Ext n+1 R (I, I) do not vanish simultaneously for any n ≥ 1.

2011

The goal of this dissertation is to provide noncommutative generalizations of the following theorems from commutative algebra: (Cohen's Theorem) every ideal of a commutative ring R is finitely generated if and only if every prime ideal of R is finitely generated, and (Kaplansky's Theorems) every ideal of R is principal if and only if every prime ideal of R is principal, if and only if R is noetherian and every maximal ideal of R is principal. We approach this problem by introducing certain families of right ideals in noncommutative rings, called right Oka families, generalizing previous work on commutative rings by T. Y. Lam and the author. As in the commutative case, we prove that the right Oka families in a ring R correspond bijectively to the classes of cyclic right R-modules that are closed under extensions. We define completely prime right ideals and prove the Completely Prime Ideal Principle, which states that a right ideal maximal in the complement of a right Oka family is completely prime. We exploit the connection with cyclic modules to provide many examples of right Oka families. Our methods produce some new results that generalize well-known facts from commutative algebra, and they also recover earlier theorems stating that certain noncommutative rings are domains-namely, proper right PCI rings and rings with the right restricted minimum condition that are not right artinian.

2012

In this paper we study (non-commutative) rings $R$ over which every finitely generated left module is a direct sum of cyclic modules (called left FGC-rings). The commutative case was a well-lnown problem studied and solved in 1970s by various authors. The main result of this paper shows that a Noetherian local left FGC-ring is either an Artinian principal left ideal ring, or an Artinian principal right ideal ring, or a prime ring over which every two-sided ideal is principal as a left and a right ideal. As a consequence, we obtain that if $R=\Pi_{i=1}^n R_i$ is a finite product of Noetherian duo-rings $R_i$ where each $R_i$ is prime or local, then $R$ is a left FGC-ring if and only if $R$ is a principal ideal ring.

2018

We introduce a weakly symmetric ring which is a generalization of a symmetric ring and a strengthening of both a GWS ring and a weakly reversible ring, and investigate properties of the class of this kind of rings. A ring R is called weakly symmetric if for any a, b, c ∈ R, abc being nilpotent implies that Racrb is a nil left ideal of R for each r ∈ R. Examples are given to show that weakly symmetric rings need to be neither semicommutative nor symmetric. It is proved that the class of weakly symmetric rings lies also between those of 2-primal rings and directly finite rings. We show that for a nil ideal I of a ring R, R is weakly symmetric if and only if R/I is weakly symmetric. If R[x] is weakly symmetric, then R is weakly symmetric, and R[x] is weakly symmetric if and only if R[x;x−1] is weakly symmetric. We prove that a weakly symmetric ring which satisfies Köthe’s conjecture is exactly an NI ring. We also deal with some extensions of weakly symmetric rings such as a Nagata exte...

2009

Let R be a commutative ring with 1 6D 0 and Nil.R/ be its set of nilpotent elements. Recall that a prime ideal of R is called a divided prime if P.x/ for every x 2 RnP ; thus a divided prime ideal is comparable to every ideal ofR. In many articles, the author investigated the class of rings HD"RjR is a commutative ring and Nil.R/ is a divided prime ideal ofR" (Observe that ifR is an integral domain, thenR2 H.) IfR2 H , thenR is called a -ring. Recently, David Anderson and the author generalized the concept of PrR ufer domains, Bezout domains, Dedekind domains, and Krull domains to the context of rings that are in the class H. Also, Lucas and the author generalized the concept of Mori domains to the context of rings that are in the class H. In this paper, we state many of the main results on -rings.

Journal of the Australian Mathematical Society, 1992

Let M be a Γ-ring with right operator ring R. We define one-sided ideals of M and show that there is a one-to-one correspondence between the prime left ideals of M and R and hence that the prime radical of M is the intersection of its prime left ideals. It is shown that if M has left and right unities, then M is left Noetherian if and only if every prime left ideal of M is finitely generated, thus extending a result of Michler for rings to Γ-rings.Bi-ideals and quasi-ideals of M are defined, and their relationships with corresponding structures in R are established. Analogies of various results for rings are obtained for Γ-rings. In particular we show that M is regular if and only if every bi-ideal of M is semi-prime.

2009

Completely prime right ideals are introduced as a one-sided generalization of the concept of a prime ideal in a commutative ring. Some of their basic properties are investigated, pointing out both similarities and differences between these right ideals and their commutative counterparts. We prove the Completely Prime Ideal Principle, a theorem stating that right ideals that are maximal in a specific sense must be completely prime. We offer a number of applications of the Completely Prime Ideal Principle arising from many diverse concepts in rings and modules. These applications show how completely prime right ideals control the one-sided structure of a ring, and they recover earlier theorems stating that certain noncommutative rings are domains (namely, proper right PCI rings and rings with the right restricted minimum condition that are not right artinian). In order to provide a deeper understanding of the set of completely prime right ideals in a general ring, we study the special subset of comonoform right ideals.

Communications in Algebra, 2015

We study certain (two-sided) nil ideals and nilpotent ideals in a Lie nilpotent ring R. Our results lead us to showing that the prime radical rad(R) of R comprises the nilpotent elements of R, and that if L is a left ideal of R, then L + rad(R) is a two-sided ideal of R. This in turn leads to a Lie nilpotent version of Cohen's theorem, namely if R is a Lie nilpotent ring and every prime (two-sided) ideal of R is finitely generated as a left ideal, then every left ideal of R containing the prime radical of R is finitely generated (as a left ideal). For an arbitrary ring R with identity we also consider its so-called n-th Lie center Zn(R), n ≥ 1, which is a Lie nilpotent ring of index n. We prove that if C is a commutative submonoid of the multiplicative monoid of R, then the subring Zn(R) ∪ C of R generated by the subset Zn(R) ∪ C of R is also Lie nilpotent of index n.