CHARACTERIZING THE NIL RADICAL OF A GAMMA RING

ABSTRACT

The nil radical, N(M) of a Γ-ring M was defined by Coppage and Luh [3], and shown by Groenewald [4] to be a special radical. We define s-prime ideals of M and show that N(M) is equal to the intersection of the s-prime ideals of M. If R is a ring, the nil radical of R considered as a Γ-ring with Γ = R is equal to the upper nil radical of R. We also give a sufficient condition for the equality N(R)* = N(M), where R is the right operator ring of M, and N(R) is its upper nil radical.     

NOTES ON NILPOTENCY OF NIL SUBRINGS OF ENDOMORPHISM RINGS OF MODULES

Abstract

A family K of right R-modules is called a natural class if K is closed under submodules, direct sums, infective hulls, and isomorphic copies. The main result of this note is the following: Let K be a natural class on Mod-R and M ε K. If M satisfies a.c.c. (or d.c.c.) on the set of submodules {N ? M: M/N ε K}, then each nil subring of End(M_R ) is nilpotent.     

BOUNDS FOR INDICES OF NILPOTENCY OF FINITELY GENERATED NIL RINGS OF FINITE INDEX

Abstract

An improved bound is given for the index of nil-potency of a finitely generated nil ring of index n in terms of the index of nilpotency of the ideal generated by T^m where m = [n/2] and T is a m-subset of the set of generators. If m = 3 it is proved that T¹⁰ is contained in an ideal generated by twenty-seven cubes and this is applied to get bounds for the index of nilpotency of a finitely generated nil ring of index 6 or 7, bounds which are less than one hundredth of the bounds we obtained in a previous paper.     

A NOTE ON LEFT STROUGLY NIL AND LEFT STROUGLY NILPOTENT

Abstract

In [2] van der Walt called a left ideal L of a ring A, left strongly nil, if given 1 ε L and k ε K, K a left ideal. there is an n such that (1+k)ⁿ ε K. L is called left strongly nilpotent if for any left ideal K there exists an m such that (L+K)^m ? K. In this paper we will prove that if A is a left artinian ring (not necessarily with unity) then every left strongly nil left ideal is left strongly nilpotent. This result is a generalization of the main theorem of [2].     

Generating dual Baer modules via fully invariant submodules

Abstract

In this paper, we introduce a concept of a dual F-Baer module M where F is the fully invariant submodule of M, by this means we deal with generating dual Baer modules. We investigate direct sums of dual F -Baer modules M by exerting the notion of relatively dual F-Baer modules. We also obtain applications of dual F-Baer modules to rings and the preradical Z*(·).     

On the Schröder–Bernstein property for modules

The well known Schröder–Bernstein Theorem states that any two sets with one to one maps into each other are isomorphic. The question of whether any two (subisomorphic or) direct summand subisomorphic algebraic structures are isomorphic, has long been of interest. Kaplansky asked whether direct summands subisomorphic abelian groups are always isomorphic? The question generated a great deal of interest. The study of this question for the general class of modules has been somewhat limited. We extend the study of this question for modules in this paper. We say that a module Msatisfies the Schröder–Bernstein property (S-B property) if any two direct summands of M which are subisomorphic to direct summands of each other, are isomorphic. We show that a large number of classes of modules satisfy the S-B property. These include the classes of quasi-continuous, directly finite, quasi-discrete and modules with ACC on direct summands. It is also shown that over a Noetherian ring R, every extending module satisfies the S-B property. Among applications, it is proved that the class of rings R for which every R-module satisfies the S-B property is precisely that of pure-semisimple rings. We show that over a commutative domain R, any two quasi-continuous subisomorphic R-modules are isomorphic if and only if R is a PID. We study other conditions related to the S-B property and obtain characterizations of certain classes of rings via those conditions. Examples which delimit and illustrate our results are provided.     

The cancellable range of rings

We unify the cancellation property of rings with stable range one and the principal ideal domain by introducing a new notion which is called “cancellable range”. It is proved that if a ring R has cancellable range n for some positive integer n, then for any n-generated module B and any module implies B ≅ C; if R is a Noetherian ring and R has cancellable range n for any n ≧ 1, then R has the cancellation property. Received: 16 November 2004     

Projective modules are determined by their radical factors

We show that two infinitely generated projective modules are isomorphic whenever they have isomorphic factors modulo their Jacobson radical. Some applications of the result to semilocal rings with indecomposable non-finitely generated projective modules are given.     

Quasi-Baer module hulls and applications

Let V be a module with $S=\mathrm{End}(V)$. V is called a quasi-Baer module if for each ideal J of S, $r_V(J)=eV$ for some $e^2=e\in S$. On the other hand, V is called a Rickart module if for each $?\in S$, $\mathrm{Ker}(?)=eV$ for some $e^2=e\in S$. For a module N, the quasi-Baer module hull $\mathbf{qB}(N)$ (resp., the Rickart module hull $\mathbf{Ric}(N)$) of N, if it exists, is the smallest quasi-Baer (resp., Rickart) overmodule, in a fixed injective hull $E(N)$ of N. In this paper, we initiate the study of quasi-Baer and Rickart module hulls. When a ring R is semiprime and ideal intrinsic over its center, it is shown that every finitely generated projective R-module has a quasi-Baer hull. Let R be a Dedekind domain with F its field of fractions and let $\{K_i|i\in \mathrm{\Lambda }\}$ be any set of R-submodules of $F_R$. For an R-module $M_R$ with ${\mathrm{Ann}}_R(M)\ne 0$, we show that $M_R\oplus {({?}_{i\in \mathrm{\Lambda }}{K}_i)}_R$ has a quasi-Baer module hull if and only if $M_R$ is semisimple. This quasi-Baer hull is explicitly described. An example such that $M_R\oplus {({?}_{i\in \mathrm{\Lambda }}{K}_i)}_R$ has no Rickart module hull is constructed. If N is a module over a Dedekind domain for which $N/t(N)$ is projective and ${\mathrm{Ann}}_R(t(N))\ne 0$, where $t(N)$ is the torsion submodule of N, we show that the quasi-Baer hull $\mathbf{qB}(N)$ of N exists if and only if $t(N)$ is semisimple. We prove that the Rickart module hull also exists for such modules N. Furthermore, we provide explicit constructions of $\mathbf{qB}(N)$ and $\mathbf{Ric}(N)$ and show that in this situation these two hulls coincide. Among applications, it is shown that if N is a finitely generated module over a Dedekind domain, then N is quasi-Baer if and only if N is Rickart if and only if N is Baer if and only if N is semisimple or torsion-free. For a direct sum $N_R$ of finitely generated modules, where R is a Dedekind domain, we show that N is quasi-Baer if and only if N is Rickart if and only if N is semisimple or torsion-free. Examples exhibiting differences between the notions of a Baer hull, a quasi-Baer hull, and a Rickart hull of a module are presented. Various explicit examples illustrating our results are constructed.     

The algebra of one-sided inverses of a polynomial algebra

We study in detail the algebra S_n in the title which is an algebra obtained from a polynomial algebra P_n in n variables by adding commuting, left (but not two-sided) inverses of the canonical generators of P_n. The algebra S_n is non-commutative and neither left nor right Noetherian but the set of its ideals satisfies the a.c.c., and the ideals commute. It is proved that the classical Krull dimension of S_n is 2n; but the weak and the global dimensions of S_n are n. The prime and maximal spectra of S_n are found, and the simple S_n-modules are classified. It is proved that the algebra S_n is central, prime, and catenary. The set I_n of idempotent ideals of S_n is found explicitly. The set I_n is a finite distributive lattice and the number of elements in the set I_n is equal to the Dedekind number d_n.     

QUASIREGULAR TORSION RINGS HAVING ISOMORPHIC ADDITIVE AND CIRCLE COMPOSITION GROUPS

Abstract

We investigate the role played by torsion properties in determining whether or not a commutative quasiregular ring has its additive and circle composition (or adjoint) groups isomorphic. We clarify and extend some results for nil rings, showing, in particular, that an arbitrary torsion nil ring has the isomorphic groups property if and only if the components from its primary decomposition into p-rings do too.

We look at the more specific case of finite rings, extending the work of others to show that a non-trivial ring with the isomorphic groups property can be constructed if the additive group has one of the following groups in its decomposition into cyclic groups: Z₂ ⁿ (for n ≥ 3), Z₂ ⊕ Z₂ ⊕ Z₂, Z₂ ⊕ Z₄, Z₄ ⊕ Z₄, Z _p ⊕ Z _p (for odd primes, p), or Z _p ⁿ (for odd primes, p, and n ≥ 2).

We consider, also, an example of a ring constructed on an infinite torsion group and use a specific case of this to show that the isomorphic groups property is not hereditary.     

The Nil Radicals in Several Classes of Lattice-Ordered Rings

We characterize the nil radicals in some l-rings. Several sufficient conditions for the nil radicals of l-rings to be equal and nilpotent are given. The results improve corresponding ones of Birkhoff and Pierce. We also show that the ideal generated by a nil one-sided ideal which is contained in a nil one-sided l-ideal is nil.     

Preinjective modules over pure semisimple rings

For a left pure semisimple ring R, it is shown that the local duality establishes a bijection between the preinjective left R-modules and the preprojective right R-modules, and any preinjective left R-module is the source of a left almost split morphism. Moreover, if there are no nonzero homomorphisms from preinjective modules to non-preinjective indecomposable modules in R-mod, the direct sum of all non-preinjective indecomposable direct summands of products of preinjective left R-modules is a finitely generated product-complete module. This generalizes a recent theorem of Angeleri Hügel [L. Angeleri Hügel, A key module over pure-semisimple hereditary rings, J. Algebra 307 (2007) 361-376] for hereditary rings.     

Relative copure injective and copure flat modules

Let R be a ring, n a fixed nonnegative integer and I_n (F_n) the class of all left (right) R-modules of injective (flat) dimension at most n. A left R-module M (resp., right R-module F) is called n-copure injective (resp., n-copure flat) if (resp., ) for any N∈I_n. It is shown that a left R-module M over any ring R is n-copure injective if and only if M is a kernel of an I_n-precover f:A→B of a left R-module B with A injective. For a left coherent ring R, it is proven that every right R-module has an F_n-preenvelope, and a finitely presented right R-module M is n-copure flat if and only if M is a cokernel of an F_n-preenvelope K→F of a right R-module K with F flat. These classes of modules are also used to construct cotorsion theories and to characterize the global dimension of a ring under suitable conditions.     

p.p. rings and generalized p.p. rings

This paper concerns two conditions, called right p.p. and generalized right p.p., which are generalizations of Baer rings and von Neumann regular rings. We study the subrings and extensions of them, adding proper examples and counterexamples to some situations and questions that occur naturally in the process of this paper.     

Sigma-cotorsion rings

It is proved that a ring R is right perfect if and only if it is Σ-cotorsion as a right module over itself. Several other conditions are shown to be equivalent. For example, that every pure submodule of a free right R-module is strongly pure-essential in a direct summand, or that the countable direct sum of the cotorsion envelope of R_R is cotorsion.If C_R is a flat Σ-cotorsion module, then C_R admits a decomposition into a direct sum of indecomposable modules with a local endomorphism ring. The Jacobson radical J(S) of the endomorphism ring S=End_RC is characterized as the maximum ideal that acts locally T-nilpotently on C_R. If R is semilocal and C=C(R), then the radical consists of those endomorphisms whose image is contained in CJ.