Morphic rings and unit regular rings

A ring R is called left morphic if for every a∈R. A left and right morphic ring is called a morphic ring. If M_n(R) is morphic for all n≥1 then R is called a strongly morphic ring. A well-known result of Erlich says that a ring R is unit regular iff it is both (von Neumann) regular and left morphic. A new connection between morphic rings and unit regular rings is proved here: a ring R is unit regular iff R[x]/(xⁿ) is strongly morphic for all n≥1 iff R[x]/(x²) is morphic. Various new families of left morphic or strongly morphic rings are constructed as extensions of unit regular rings and of principal ideal domains. This places some known examples in a broader context and answers some existing questions.     

Morphic group rings

An element a in a ring R is called left morphic if there exists b∈R such that l_R(a)=Rb and l_R(b)=Ra, where l_R(a) denotes the left annihilator of a in R. The ring R is called left morphic if every element of R is left morphic. Left morphic rings have been studied by Nicholson and Sánchez Campos. In this paper, the question of when a group ring is left morphic is discussed in great detail and various morphic group rings are identified.     

Coherence and Generalized Morphic Property of Triangular Matrix Rings

Let R be a ring. R is left coherent if each of its finitely generated left ideals is finitely presented. R is called left generalized morphic if for every element a in R, l(a) = Rb for some b ∈ R, where l(a) denotes the left annihilator of a in R. The main aim of this article is to investigate the coherence and the generalized morphic property of the upper triangular matrix ring T _n (R) (n ≥ 1). It is shown that R is left coherent if and only if T _n (R) is left coherent for each n ≥ 1 if and only if T _n (R) is left coherent for some n ≥ 1. And an equivalent condition is obtained for T _n (R) to be left generalized morphic. Moreover, it is proved that R is left coherent and left Bézout if and only if T _n (R) is left generalized morphic for each n ≥ 1.     

Characterizations of QF Rings

A ring R is called “left GP-injective” if for any 0 ≠ a ∈ R, there exists n > 0 such that a ⁿ  ≠ 0 and a ⁿ R = r(l(a ⁿ )). It is proved that (1) every right Noetherian left GP-injective ring such that every complement left ideal is a left annihilator is a QF ring, (2) every left GP-injective ring with ACC on left annihilators such that every complement left ideal is a left annihilator is a QF ring, and (3) every left P-injective left CS ring satisfying ACC on essential right ideals is a QF ring. Several well-known results on QF rings are obtained as corollaries.     

FP-RINGS

A ring R is called right FP-injective if every R-homomor-phism from a finitely generated submodule of a free right R-module F into R extends to F. In this paper a ring R will be called a right FP-ring if R is semiperfect, right FP-injective and has an essential right socle. The class of FP-rings strictly contains the class of right (and left) pseudo-Frobenius rings, and we show that it is right-left symmetric and Morita-invariant. As an application we show that if R is a left perfect right FP-injective ring, then R is quasi-Frobenius if and only if the second right socle of R is finitely generated as a right ideal of R. This extends the known results in the right selfinjective case.     

Rings Whose Modules Have Grade Zero

In this paper, we prove that R is a two-sided Artinian ring and J is a right annihilator ideal if and only if (i) for any nonzero right module, there is a nonzero linear map from it to a projective module; (ii) every submodule of RR is not a radical module for some right coherent rings. We call a ring a right X ring if Homa(M, R) = 0 for any right module M implies that M = 0. We can prove some left Goldie and right X rings are right Artinian rings. Moreover we characterize semisimple rings by using X rings. A famous Faith‘s conjecture is whether a semipimary PF ring is a QF ring. Similarly we study the relationship between X rings and QF and get many interesting results.     

𝔏-Rickart Modules

It is well known that the Rickart property of rings is not a left-right symmetric property. We extend the notion of the left Rickart property of rings to a general module theoretic setting and define 𝔏-Rickart modules. We study this notion for a right R-module M _R where R is any ring and obtain its basic properties. While it is known that the endomorphism ring of a Rickart module is a right Rickart ring, we show that the endomorphism ring of an 𝔏-Rickart module is not a left Rickart ring in general. If M _R is a finitely generated 𝔏-Rickart module, we prove that End _R (M) is a left Rickart ring. We prove that an 𝔏-Rickart module with no set of infinitely many nonzero orthogonal idempotents in its endomorphism ring is a Baer module. 𝔏-Rickart modules are shown to satisfy a certain kind of nonsingularity which we term “endo-nonsingularity.” Among other results, we prove that M is endo-nonsingular and End _R (M) is a left extending ring iff M is a Baer module and End _R (M) is left cononsingular.     

On Skew Triangular Matrix Rings

For a ring R, endomorphism α of R and positive integer n we define a skew triangular matrix ring T _n (R, α). By using an ideal theory of a skew triangular matrix ring T _n (R, α) we can determine prime, primitive, maximal ideals and radicals of the ring R[x; α]/ ? x ⁿ  ?, for each positive integer n, where R[x; α] is the skew polynomial ring, and ? x ⁿ  ? is the ideal generated by x ⁿ .     

ALMOST COHEN–MACAULAY MODULES

A commutative noetherian ring R is called an almost Cohen–Macaulay ring if depth(P, R) = depth(P R _P , R _P ) for every P ∈ SpecR. [It was called a D-ring by Han (Acta Math. Sinica 1998, 4, 1047–1052).] Several fundamental properties of almost Cohen–Macaulay rings were estabished by Han. In this note, a new characterization is proved: R is an almost Cohen–Macaulay ring if and only if height P ≤ 1 + depth(P, R) for every P ∈ Spec(R). By this characterization, we settle an unsolved problem in Han's paper: R is an almost Cohen–Macaulay ring if and only if so is the power series ring R[[X ₁, …, X _n ]]. The notion of an almost Cohen–Macaulay ring is generalized to that of an almost Cohen–Macaulay module in this note.     

Study of Hopficity in Certain Classes of Rings

The main results proved in this paper are:

1. For any non-zero vector space V _Dover a division ring D, the ring R= End(V _D) is hopfian as a ring

2. Let Rbe a reduced π-regular ring &; B(R) the boolean ring of idempotents of R. If B(R) is hopfian so is R.The converse is not true even when Ris strongly regular.

3. Let Xbe a completely regular spaceC(X) (resp. C ^?(X)) the ring of real valued (resp. bounded real valued) continuous functions on X. Let Rbe any one of C(X) or C ^?(X). Then Ris an exchange ring if &; only if Xis zero dimensional in the sense of Katetov. for any infinite compact totally disconnected space X C(X) is an exchange ring which is not von Neumann regular.

4. Let Rbe a reduced commutative exchange ring. If Ris hopfian so is the polynomial ring R[T ₁,…,T _n] in ncommuting indeterminates over Rwhere nis any integer ≥ 1.

5. Let Rbe a reduced exchange ring. If Ris hopfian so is the polynomial ring R[T].     

Some Characterizations of VNL Rings

A ring R is said to be von Newmann local (VNL) if for any a ∈ R, either a or 1 ?a is (von Neumann) regular. The class of VNL rings lies properly between exchange rings and (von Neumann) regular rings. We characterize abelian VNL rings. We also characterize and classify arbitrary VNL rings without an infinite set of orthogonal idempotents; and also the VNL rings having a primitive idempotent e such that eRe is not a division ring. We prove that a semiperfect ring R is VNL if and only if for any right uni-modular row (a ₁, a ₂) ∈ R ², one of the a _i 's is regular in R. Formal triangular matrix rings that are VNL are also characterized. As a corollary, it is shown that an upper triangular matrix ring T _n (R) is VNL if and only if n = 2 or 3 and R is a division ring.     

Generalizations of coatomic modules

For a ring R and a right R-module M, a submodule N of M is said to be δ-small in M if, whenever N+X=M with M/X singular, we have X=M. Let ℘ be the class of all singular simple modules. Then δ(M)=Σ{ L≤ M| L is a δ-small submodule of M} = Re _jm(℘)=∩{ N⊂ M: M/N∈℘. We call M δ-coatomic module whenever N≤ M and M/N=δ(M/N) then M/N=0. And R is called right (left) δ-coatomic ring if the right (left) R-module R _R(_RR) is δ-coatomic. In this note, we study δ-coatomic modules and ring. We prove M=⊕ _i=1 ⁿ M_i is δ-coatomic if and only if each M _i (i=1,…, n) is δ-coatomic.     

Mod-Retractable Rings

A right module M over a ring R is said to be retractable if Hom _R (M, N) ≠ 0 for each nonzero submodule N of M. We show that M ? _R RG is a retractable RG-module if and only if M _R is retractable for every finite group G. The ring R is (finitely) mod-retractable if every (finitely generated) right R-module is retractable. Some comparisons between max rings, semiartinian rings, perfect rings, noetherian rings, nonsingular rings, and mod-retractable rings are investigated. In particular, we prove ring-theoretical criteria of right mod-retractability for classes of all commutative, left perfect, and right noetherian rings.     

On finitely embedded rings

A result of Ginn and Moss asserts that a left and right noetherian ring with essential right socle is left and right artinian. There are examples of right finitely embedded rings with ACC on left and right annihilators which are not artinian. Motivated by this, it was shown by Faith that a commutative, finitely embedded ring with ACC on annihilators (and square-free socle) is artinian (quasi-Frobenius). A ring R is called right minsymmetric if, whenever k R is a simple right ideal of R, then R k is also simple. In this paper we show that a right noetherian right minsymmetric ring with essential right socle is right artinian. As a consequence we show that a ring is quasi-Frobenius if and only if it is a right and left mininjective, right finitely embedded ring with ACC on right annihilators. This extends the known work in the artinian case, and also extends Faith's result to the non-commutative case.     

The homomorphismM*⊗N → Hom (M,N)

In this note we prove two theorems. In theorem 1 we prove that if M andN are two non-zero reflexive modules of finite projective dimensions over a Gorenstein local ring, such that Hom (M, N) is a third module of syzygies, then the natural homomorphismM* ⊗N → Hom (M, N) is an isomorphism. This extends the result in [7]. In theorem 2, we prove that projective dimension of a moduleM over a regular local ringR is less than or equal ton if and only if Ext_R ⁿ (M, R) ⊗M → Ext_R ⁿ (M, M) is surjective; in which case it is actually bijective. This extends the usual criterion for the projectivity of a module.     

PI-rings and semiprime rings having faithful modules with krull dimension

It is proved that if a PI-ring R has a faithful left R-module M with Krull dimension, then its prime radical rad(R) is nilpotent. If, moreover, the R-module M and the left ideal_R(rad(R)) are finitely generated, then R has a left Krull dimension equal to the Krull dimension of M. It turns out that a semiprime ring, which has a faithful (left or right) module with Krull dimension, is a finite subdirect product of prime rings. Moreover, first, a right Artinian ring R such that rad(R)²=0 has a faithful Artinian cyclic left module, and second, a finitely generated semiprime PI-algebra over a field has a faithful Artinian module. We give examples showing that the restrictions imposed are essential, as well as an example of a finitely generated prime PI-algebra over a field, which is not Noetherian and has a Krull dimension. Supported by RFFR grant No. 26-93-011-1544. Translated fromAlgebra i Logika, Vol. 36, No. 5, pp. 562–572, September–October, 1997.     

Left SF-Rings and Regular Rings

A ring R is called a left (right) SF-ring if all simple left (right) R-modules are flat. It is known that von Neumann regular rings are left and right SF-rings. In this article, we study the regularity of left SF-rings and we prove the following: 1) if R is a left SF-ring whose all complement left (right) ideals are W-ideals, then R is strongly regular; 2) if R is a left SF-ring whose all maximal essential right ideals are GW-ideals, then R is regular.     

Orders in regular rings with minimal condition for principal right ideals

ABSTRACT

A ring R is called an n-clean (resp. Σ-clean) ring if every element in R is n-clean (resp. Σ-clean). Clean rings are 1-clean and hence are Σ-clean. An example shows that there exists a 2-clean ring that is not clean. This shows that Σ-clean rings are a proper generalization of clean rings. The group ring ?_(p) G with G a cyclic group of order 3 is proved to be Σ-clean. The m× m matrix ring M _m (R) over an n-clean ring is n-clean, and the m×m (m>1) matrix ring M _m (R) over any ring is Σ-clean. Additionally, rings satisfying a weakly unit 1-stable range were introduced. Rings satisfying weakly unit 1-stable range are left-right symmetric and are generalizations of abelian π-regular rings, abelian clean rings, and rings satisfying unit 1-stable range. A ring R satisfies a weakly unit 1-stable range if and only if whenever a ₁ R + ˙˙˙ a _m R = dR, with m ≥ 2, a ₁,…, a _m, d ∈ R, there exist u ₁ ∈ U(R) and u ₂,…, u _m ∈ W(R) such that a ₁ u ₁ + ? a _m u _m = Rd.