COPURE INJECTIVE MODULES

Abstract

A module is said to be copure injective if it is injective with respect to all modules A ? B with B/A injective. We first characterize submodules that have the extension property with respect to copure injective modules. Then we characterize commutative rings with finite self injective dimension in terms of copure injective modules. Finally, we show that the quotient categories of reduced copure injective modules and reduced h- divisible modules are isomorphic.     

THE LARGEST NON-TRIVIAL TORSION CLASS OF MODULES

ABSTRACT

In this paper we investigate the following two classes of left R-modules: N(P) ={A|A has no non-zero direct summand P ε P} and H(p) = {A} if B ? A with B ε N(P), then B = 0}, where P is a class of projective R-modules. We demonstrate that N(p) is, in general, not a torsion class but that H(P) is always a torsionfree class. We also investigate those classes P and rings R for which N(P) is the largest non-trivial torsion class of R-modules.     

ENDOMORPHISMS OF VERMA MODULES FOR RATIONAL CHEREDNIK ALGEBRAS

We study the endomorphism algebras of Verma modules for rational Cherednik algebras at t = 0. It is shown that, in many cases, these endomorphism algebras are quotients of the centre of the rational Cherednik algebra. Geometrically, they define Lagrangian subvarieties of the generalized Calogero–Moser space. In the introduction, we motivate our results by describing them in the context of derived intersections of Lagrangians.     

MODULO-CONSTANT IDEAL-HEREDITARY RADICALS OF NEAR-KINGS

ABSTRACT

It is known that no “good” radical of (not necessarily o-symmetric) near-rings can be ideal-hereditary. Using the results of the o-symmetric case, we show that the situation is not as bad as on first appearances and we give several examples of (Kurosh-Amitsur) radicals of near-rings for which the semisimple class is hereditary and the radical class is hereditary on left invariant ideals. We also extend some recent results on left strong radicals from the o-symmetric case to the general case.     

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.     

COTORSION MODULES AND PROJECTIVE DIMENSION ONE

If T is a perfect torsion theory for a category of modules over a commutative ring R, a module C is called T—cotorsion provided Hom_R(Q_T,C) = 0 = Ext_R (Q_T,C) where Q_T denotes the T-injective hull of R. Motivated by the now classical results of D. K. Harrison for abelian groups and of E. Matlis for modules over a domain, the theory of T—cotorsion modules is extended. For example, a category equivalence is obtained between the category of T—compact T-cotorsion modules and the category of T-torsion T-reduced modules. The class of T-divisible modules (homomorphic images of T-injective modules) is shown to be closed under formation of extensions if and only if pd_RQ_T ≤ 1, in the case that Q_T is T—cocritical.     

PI—环上有限生成模的自同态的一个注记

本文将交换环上有限生成模的单自同态的有关结果推广到ＰＩ－环上,得到如下类似结果。     

交换环上矩阵保逆的模自同态

本文刻划交换环上全矩阵模保矩阵逆的自同态。     

SPECIAL RADICALS OF NEAR-RING MODULES

Abstract

Equiprime near-rings, which generalize the concept of prime-ness in rings, were defined by the present authors, together with S. Veldsman. This concept was shown in subsequent work to lead to a very satisfactory theory of special radicals for near-rings. In the current paper, we define equiprime N-groups for a near-ring N. It is shown that an ideal A of N is equiprime if and only if it is the annihilator of an equiprime TV-group G. Special classes of near-ring modules are defined, and a module-theoretic characterization of special radicals of near-rings is established, similar to that given by Andrunakievich and Rjabuhin for special radicals of rings.     

A characterization of prime radical in near-rings under chain conditions on annihilators

It has been proved that ifR is a near-ring that satisfiesd.c.c anda.c.c. on right annihilators of its righR-subsets, then the prime radicalP(r) is a nilpotent ideal. A few results are included in Author's Doctoral Dissertation at Sukhadia University, Udaipur (1983).     

STRONGLY EQUIPRIME NEAR-RINGS

Abstract

Strongly equiprime near-rings are defined which generalize strongly prime rings to near-rings. These near-rings determine an ideal-hereditary Kurosh-Amitsur radical in the variety of 0-symmetric near-rings. In the same variety, the uniformly strongly equiprime near-rings also determine an ideal-hereditary Kurosh-Amitsur radical which is not comparable with the Jacobson-type radicals nor with the Brown-McCoy-type radicals.     

PRIMITIVITY IN MATRIX NEAR-RINGS

Abstract

The relationships between modules of a near-ring R and the matrix near-ring IM_n(R) are studied, especially as regards primitivity. It is shown that R is 2-primitive iff IM_n(R) is.     

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.     

EMBEDDING MODULES IN GRADED MODULES OVER A SEMIGROUP-GRADED RING

We analyze conditions under which an (ungraded) module over a semigroup-graded ring can be embedded in a graded module. We use this information to provide some sufficient conditions under which the graded Jacobson radical of the ring is contained in the (ungraded) Jacobson radical. These include, and give an alternate approach to, the case of group-graded rings.

     

EQUIPRIME Γ-NEAR-RINGS

Abstract

Equiprime and strongly equiprime near-rings were recently defined by the present authors, together with S. Veldsman. In the present paper, the concepts are introduced for Γ-near-rings, and give rise to Kurosh- Amitsur radicals. If M is a Γ-near-ring and L is its left operator near-ring, then R(L)⁺ = R(M), where R(—) in both cases denotes either the equiprime or the strongly equiprime radical.     

Near-rings with no non-zero nilpotent two-sidedR-subsets

It has been proved that, ifR is a near-ring with no non-zero nilpotent two-sidedR-subsets and if the annihilator of any non-zero ideal is contained in some maximal annihilator, thenR is a subdirect sum of strictly prime near-rings. Moreover, ifR is a near-ring with no non-zero nilpotent two-sidedR-subsets and satisfying a.c.c. or d.c.c. on annihilating ideals of the form Ann (Q), whereQ is an ideal ofR, thenR is a finite subdirect sum of strictly prime near-rings. It is also proved that, ifR is a regular and right duo near-ring that satisfies a.c.c. (or d.c.c.) on annihilating ideals of the form Ann (Q), whereQ is an ideal ofR, thenR is a finite direct sum of near-ringsR _i (1 i n) where eachR _i is simple and strictly prime.     

HOMOTOPY SELF-EQUIVALENCE GROUPS OF UNIONS OF SPACES: INCLUDING MOORE-SPACES

Abstract

The group ?(M^m(A) v Mⁿ(π)) of homotopy self-equivalence classes of two Moore spaces is faithfully represented onto a (multiplicative) group of matrices for n≥m≥3. We consider, in this note, related representations of ?(M^m(Λ)vMⁿ(π)), for finitely generated Λ and π in the case where n≥4, and also where n=3 if ext(Λ, π)=0. The representation onto a matrix group, similar to that in the case above, is not, in general, valid. We show however that ?(M²(Λ)vMⁿ(π)) is represented onto ?(M²(Λ))× ?(Mⁿ(π) in this case, and that this representation determines an isomorphism with an iterated semi-direct product ?(M²(Λ)v Mⁿ(π)) ? {(Mⁿ(π), M²(Λ))? ext(π Λ ? π)} ? (?(M²(Λ)) × ? (Mⁿ(π)).

More generally we review, and-extend, the theory of the representation of the (generalized) near ring (XvY,XvY) onto the matrix (generalized) near-ring (XvY, XxY) where appropriate, in the case where X and Y are h-coloops; and we deduce results for the representation of ?(XvY, XvY). Some of the results published previously in the case of simply-connected CW co-h-spaces, extend to the case where X and Y are path-connected h-coloops one of which is well-pointed. We note the obstructions to the existence of a homomorphic section, and consider a number of special cases which occur when some of the groups are trivial.     

Notes on Brown-McCoy-radicals for Γ-near-rings

The Brown-McCoy radical is known to be an ideal-hereditary Kurosh-Amitsur radical in the variety of zerosymmetric near-rings. We define the Brown-McCoy and simplical radicals, and , respectively, for zerosymmetric -near-rings. Both and are ideal-hereditary Kurosh-Amitsur radicals in that variety. IfM is a zerosymmetric -near-ring with left operator near-ringL, it is shown that , with equality ifM has a strong left unity. is extended to the variety of arbitrary near-rings, and and are extended to the variety of arbitrary -near-rings, in a way that they remain Kurosh-Amitsur radicals. IfN is a near-ring andA N, then , with equality ifA if left invariant.     

RADICALS IN GENERAL Γ-NEAR-RINGS

Abstract

The J ₂ and J ₃ radicals for zerosymmetric Γ-near-rings were recently defined by the author. In the present paper we define the J ₂₍₀₎ and J ₃₍₀₎ radicals for arbitrary Γ-near-rings. These radicals are sirmlar to corresponding ones which were recently defined by Veldsman for near-rings. Let M be a r-near-ring with left operator near-ring L. Then J _κ(0)(L)⁺ = J _{κ (0)} (M), k. = 2,3. If A is an ideal of M, then J _{κ (0)} (A) ? J _{κ (o)}(M) ∩ A, with equality when k = 3 and A is left invariant. J ₃₍₀₎ is a Kurosh-Amitsur radical in the variety of Γ-near-rings.     

RELATIVE INJECTIVITY OF MODULES AND EXCELLENT EXTENSIONS

Abstract

Yesl Any equation of conservation of the form ?_x{P(?_xφ, ?_tφ) = ?_t{Q(?_xφ, ?_tφ) is shown to admit an infinite-dimensional, Abellan group of symmetries that is not a prolongation symmetry group. Explicit equations are given for the determination of the generators of the Lle algebra of this Abellan symmetry group, and for the generators of Its underlying Poisson algebra.