On minimal strongly prime ideals

In this paper we give some characterizations of a ring Rwhose unique maximal nil ideal N _r (R) coincides with the set of all its nilpotent elements N(R) by using its minimal strongly prime ideals.     

Zero-Divisors,Torsion Elements,and Unions of Annihilators

Let R be a commutative ring and M be a nonzero R-module. Now Z(M) = {r ∈ R | rm = 0 for some 0 ≠ m ∈ M} is a union of prime ideals of R and T(M) = {m ∈ M | rm = 0 for some 0 ≠ r ∈ R} is a union of prime submodules of M if M ≠ T(M). We investigate representations of Z(M) and T(M) as unions of primes each of which is a union of annihilators.     

Über Die Assoziierten Primideale des Bidualen

Let (R, 𝔪) be a commutative, noetherian, local ring, E the injective hull of the residue field R/𝔪, and M ^○○ = Hom _R (Hom _R (M, E), E) the bidual of an R-module M. We investigate the elements of Ass(M ^○○) as well as those of Coatt(M) = {𝔭 ∈ Spec(R)|𝔭 = Ann _R (Ann _M (𝔭))} and provide criteria for equality in one of the two inclusions Ass(M) ? Ass(M ^○○) ? Coatt(M). If R is a Nagata ring and M a minimax module, i.e., an extension of a finitely generated R-module by an artinian R-module, we show that Ass(M ^○○) = Ass(M) ∪ {𝔭 ∈ Coatt(M)| R/𝔭 is incomplete}.     

A Note on the Associated Primes of Local Cohomology Modules

In this note we give a simple proof of the following result: Let R be a commutative Noetherian ring,  an ideal of R and M a finite R-module, if H_ ⁱ (M) has finite support for all i < n, then Ass(H_ ⁿ (M)) is finite.     

Strongly Duo Modules and Rings

An R-module M is called strongly duo if Tr(N, M) = N for every N ≤ M _R . Several equivalent conditions to being strongly duo are given. If M _R is strongly duo and reduced, then End _R (M) is a strongly regular ring and the converse is true when R is a Dedekind domain and M _R is torsion. Over certain rings, nonsingular strongly duo modules are precisely regular duo modules. If R is a Dedekind domain, then M _R is strongly duo if and only if either M ≈ R or M _R is torsion and duo. Over a commutative ring, strongly duo modules are precisely pq-injective duo modules and every projective strongly duo module is a multiplication module. A ring R is called right strongly duo if R _R is strongly duo. Strongly regular rings are precisely reduced (right) strongly duo rings. A ring R is Noetherian and all of its factor rings are right strongly duo if and only if R is a serial Artinian right duo ring.     

Unique maximal rings of functions

For several classes of groups G, we characterize when the near-ring M₀(G) of 0-preserving selfmaps on G contains a unique maximal ring. Definitive results are obtained for finite Abelian, finite nilpotent, and finite permutation groups. As an application, we determine those finite groups G such that all rings in M₀(G) are commutative.     

Sets of Elements that Pairwise Generate a Matrix Ring

In this article, we investigate the relationship between the minimum number of proper subgroups of GL(n, q) whose union is the whole GL(n, q) and the maximum number of elements that pairwise generate GL(n, q). We show that the minimum number of proper subrings of M _n (q) whose union is the whole M _n (q) is exactly the maximum number of elements that pairwise generate M _n (q).     

Localisation in rings equipped with a solvable Lie algebra action

Let k be an algebraically closed uncountable field of characteristic 0,g a finite dimensional solvable k-Lie algebraR a noetherian k-algebra on which g acts by k-derivationsU(g) the enveloping algebra of g,A=R*g the crossed product of R by U(g)P a prime ideal of A and Ω(P) the clique of P. Suppose that the prime ideals of the polynomial ring R[x] are completely prime. If R is g-hypernormal, then Ω(P) is classical. Denote by A_T the localised ring and let M be a primitive ideal of A_T Set Q=P∩R In this note, we show that if R is a strongly (R,g)-admissible integral domain and if QR_Q is generated by a regular g-centralising set of elements, then

(1)M is generated by a regular g-semi-invariant normalising set of elements of cardinald = dim (R_Q 0 + ∣X_A (P)∣

(2)d gldim(A_T ) = Kdim(A_T ) = ht(M) = ht(P).     

On Lie Nilpotent Rings and Cohen's Theorem

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 Z _n (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 ?Z _n (R) ∪ C? of R generated by the subset Z _n (R) ∪ C of R is also Lie nilpotent of index n.     

On the prime radical and Baer’s lower nilradical of modules

Let M be a left R-module. In this paper a generalization of the notion of m-system set of rings to modules is given. Then for a submodule N of M, we define := { m ε M: every m-system containing m meets N}. It is shown that is the intersection of all prime submodules of M containing N. We define rad _R (M) = . This is called Baer-McCoy radical or prime radical of M. It is shown that if M is an Artinian module over a PI-ring (or an FBN-ring) R, then M/rad _R (M) is a Noetherian R-module. Also, if M is a Noetherian module over a PI-ring (or an FBN-ring) R such that every prime submodule of M is virtually maximal, then M/rad _R (M) is an Artinian R-module. This yields if M is an Artinian module over a PI-ring R, then either rad _R (M) = M or rad _R (M) = ∩ _i=1 ⁿ for some maximal ideals of R. Also, Baer’s lower nilradical of M [denoted by Nil_* ( _R M)] is defined to be the set of all strongly nilpotent elements of M. It is shown that, for any projective R-module M, rad _R (M) = Nil_*( _R M) and, for any module M over a left Artinian ring R, rad _R (M) = Nil_*( _R M) = Rad(M) = Jac(R)M. This research was in part supported by a grant from IPM (No. 85130016). Also this work was partially supported by IUT (CEAMA). The author would like to thank the anonymous referee for a careful checking of the details and for helpful comments that improved this paper.     

Reducing system of parameters and the Cohen-Macaulay property

Let R be a local ring and let (x ₁, …, x _r) be part of a system of parameters of a finitely generated R-module M, where r < dim_R M. We will show that if (y ₁, …, y _r) is part of a reducing system of parameters of M with (y ₁, …, y _r) M = (x ₁, …, x _r) M then (x ₁, …, x _r) is already reducing. Moreover, there is such a part of a reducing system of parameters of M iff for all primes P ε Supp M ∩ V _R(x ₁, …, x _r) with dim_R R/P = dim_R M − r the localization M _P of M at P is an r-dimensional Cohen-Macaulay module over R _P. Furthermore, we will show that M is a Cohen-Macaulay module iff y _d is a non zero divisor on M/(y ₁, …, y _d−1) M, where (y ₁, …, y _d) is a reducing system of parameters of M (d:= dim_R M).     

Mathematical modeling of nilpotent subsemigroups of semigroups of contracting transformations of a Boolean

We study mathematical models of the structure of nilpotent subsemigroups of the semigroup PTD(B _n ) of partial contracting transformations of a Boolean, the semigroup TD(B _n ) of full contracting transformations of a Boolean, and the inverse semigroup ISD(B _n ) of contracting transformations of a Boolean. We propose a convenient graphical representation of the semigroups considered. For each of these semigroups, the uniqueness of its maximal nilpotent subsemigroup is proved. For PTD(B _n ) and TD(B _n ) , the capacity of a maximal nilpotent subsemigroup is calculated. For ISD(B _n ), we construct estimates for the capacity of a maximal nilpotent subsemigroup and calculate this capacity for small n. For all indicated semigroups, we describe the structure of nilelements and maximal nilpotent subsemigroups of nilpotency degree k and determine the number of elements and subsemigroups for some special cases.     

Degenerations of <Emphasis Type="Italic">k</Emphasis>[<Emphasis Type="Italic">t</Emphasis>]/(<Emphasis Type="Italic">t</Emphasis><Superscript>r</Superscript>)-Modules Giving Stratification of the Orbits

Let Λ be a finitely generated associative k-algebra where k is an algebraically closed field. For each natural number d, we have the variety of d-dimensional module structures on k^d given by the multiplication of the elements from a generating set of Λ. The general linear group Gl_d(k) acts on this variety by conjugation and the orbits under this action correspond to isomorphism classes of d-dimensional Λ-modules. For two d-dimensional Λ-modules M and N one says that M degenerates to N if the orbit corresponding to N is in the Zariski-closure of the orbit corresponding to M. Now in this situation the stabilizers of the elements in the orbit corresponding to N acts on the orbit corresponding to M. In this paper we characterize degenerations of k[t]/(t^r)-modules with the property that for each y in the orbit corresponding to N, there is an x_y in the orbit corresponding to M such that the orbit corresponding to M is the disjoint union of orbits of the x_y’s under the action of the stabilizer of y where y runs through the orbit corresponding to N. Presented by Idun ReitenMathematics Subject Classifications (2000) 14L30, 16G10.     

Homogeneity of Certain Invariant Distributions on the Lie Algebra of p-adic GLn

Let F be a non-Archimedean local field with ring of integers R and prime ideal . Suppose T is a GL _n (F)-invariant distribution on =M _n (F), the Lie algebra of GL _n (F). If T has support in the set of topologically nilpotent elements, then the restriction of T to the set of functions which are compactly supported and invariant under M _n ( ) may be expressed as a linear combination of nilpotent orbital integrals restricted to the same set of functions.     

T-Semisimple Modules and T-Semisimple Rings

We define and investigate t-semisimple modules as a generalization of semisimple modules. A module M is called t-semisimple if every submodule N contains a direct summand K of M such that K is t-essential in N. T-semisimple modules are Morita invariant and they form a strict subclass of t-extending modules. Many equivalent conditions for a module M to be t-semisimple are found. Accordingly, M is t-semisiple, if and only if, M = Z ₂(M) ⊕ S(M) (where Z ₂(M) is the Goldie torsion submodule and S(M) is the sum of nonsingular simple submodules). A ring R is called right t-semisimple if R _R is t-semisimple. Various characterizations of right t-semisimple rings are given. For some types of rings, conditions equivalent to being t-semisimple are found, and this property is investigated in terms of chain conditions.     

Finite Groups with S-permutable n-maximal Subgroups

Let G be a finite group and M _n (G) be the set of n-maximal subgroups of G, where n is an arbitrary given positive integer. Suppose that M _n (G) contains a nonidentity member and all members in M _n (G) are S-permutable in G. Then any of of the following conditions guarantees the supersolvability of G: (1) M _n (G) contains a nonidentity member whose order is not a prime; (2) all nonidentity members in M _n (G) are of prime order, and all cyclic members in M _n?1(G) of order 4 are S-permutable in G.     

Pure Submodules of Finite Direct Sums of Co-Purely Indecomposable Modules

A torsion-free module M of finite rank over a discrete valuation ring R with prime p is co-purely indecomposable if M is indecomposable and rank M = 1 + dim _R/pR (M/pM). Co-purely indecomposable modules are duals of pure finite rank submodules of the p-adic completion of R. Pure submodules of cpi-decomposable modules (finite direct sums of co-purely indecomposable modules) are characterized. Included are various examples and properties of these modules.     

Ideals,Natural Classes,and Functors

For any ring R, the set 𝒩(R) of all natural classes of R-modules is a complete Boolean lattice, which is a direct sum of two convex and complete Boolean sublattices 𝒩(R) = 𝒩 _t (R) ⊕ 𝒩 _f (R), where the last summand is the set of all nonsingular natural classes. The ring R contains a unique lattice of ideals 𝒥(R) which is lattice isomorphic to 𝒩 _f (R). The present note develops the analogue of all of the above for an arbitrary R-module M, so that in the special case when M _R  = R _R , the known lattice isomorphism 𝒥(R) ? 𝒩 _f (R) is recovered.     

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.