Relative Projective Modules and Relative Injective Modules

Let R be a ring, and n and d fixed non-negative integers. An R-module M is called (n, d)-injective if Ext ^d+1 _R (P, M) = 0 for any n-presented R-module P. M is said to be (n, d)-projective if Ext¹ _R (M, N) = 0 for any (n, d)-injective R-module N. We use these concepts to characterize n-coherent rings and (n, d)-rings. Some known results are extended.     

Sufficient Condition to Resolve Costa's First Conjecture

In this work, we give a sufficient condition to resolve Costa's first conjecture for each positive integer n and d with n ≥ 4. Precisely, we show that if there exists a local ring (A, M) such that λ _A (M) = n, and if there exists an (n + 2)-presented A-submodule of M ^m , where m is a positive integer (for instance, if M contains a regular element), then we may construct an example of (n + 4, d)-ring which is neither an (n + 3, d)-ring nor an (n + 4, d ? 1)-ring. Finally, we construct a local ring (B, M) such that λ _B (M) = 0 (resp., λ _B (M) = 1) and so we exhibit for each positive integer d, an example of a (4, d)-ring (resp., (5, d)-ring) which is neither a (4, d ? 1)-ring (resp., neither a (5, d ? 1)-ring) nor a (2, d′)-ring (resp., nor a (3, d′)-ring) for each positive integer d′.     

On Generalizations of Prime Ideals (II)

Let R be a commutative ring with identity. We say that a proper ideal P of R is (n ? 1, n)-weakly prime (n ≥ 2) if 0 ≠ a ₁…a _n  ∈ P implies a ₁…a _i?1 a _i+1…a _n  ∈ P for some i ∈ {1,…, n}, where a ₁,…, a _n  ∈ R. In this article, we study (n ? 1, n)-weakly prime ideals. A number of results concerning (n ? 1, n)-weakly prime ideals and examples of (n ? 1, n)-weakly prime ideals are given. Rings with the property that for a positive integer n such that 2 ≤ n ≤ 5, every proper ideal is (n ? 1, n)-weakly prime are characterized. Moreover, it is shown that in some rings, nonzero (n ? 1, n)-weakly prime ideals and (n ? 1, n)-prime ideals coincide.     

On Copure Projective Modules and Copure Projective Dimensions

Let R be any ring. A right R-module M is called n-copure projective if Ext¹(M, N) = 0 for any right R-module N with fd(N) ≤ n, and M is said to be strongly copure projective if Ext ⁱ (M, F) = 0 for all flat right R-modules F and all i ≥ 1. In this article, firstly, we present some general properties of n-copure projective modules and strongly copure projective modules. Then we define and investigate copure projective dimensions of modules and rings. Finally, more properties and applications of n-copure projective modules, strongly copure projective modules and copure projective dimensions are given over coherent rings with finite self-FP-injective dimension.     

On Generalizations of Prime Ideals

Let R be a commutative ring with identity. Let φ: S(R) → S(R) ∪ {?} be a function, where S(R) is the set of ideals of R. Suppose n ≥ 2 is a positive integer. A nonzero proper ideal I of R is called (n ? 1, n) ? φ-prime if, whenever a ₁, a ₂, ?, a _n  ∈ R and a ₁ a ₂?a _n  ∈ I?φ(I), the product of (n ? 1) of the a _i 's is in I. In this article, we study (n ? 1, n) ? φ-prime ideals (n ≥ 2). A number of results concerning (n ? 1, n) ? φ-prime ideals and examples of (n ? 1, n) ? φ-prime ideals are also given. Finally, rings with the property that for some φ, every proper ideal is (n ? 1, n) ? φ-prime, are characterized.     

Classes of Commutative Rings Defined by Special Conditions

ABSTRACT

In this article, we are mainly concerned with (n, d)-Krull rings, i.e., rings in which each n-presented prime ideal has height at most d. Precisely, we show that weakly n-Von Neumann regular rings are (n ? 1, 0)-Krull rings. Also, we prove that (n, d)-Krull property is not local property and that R is an (n, d)-Krull ring if and only if dim(R _P ) ≤ d for each n-presented prime ideal P of R. Finally, we construct a class of (2, d)-Krull rings which are neither (2, d ? 1)-Krull rings (for d = 1) nor (1, d)-Krull rings for d = 0,1.     

Values of the Euler Function in Various Sequences

Let ϕ(n) and λ(n) denote the Euler and Carmichael functions, respectively. In this paper, we investigate the equation ϕ(n)^r = λ(n)^s, where r ≥ s ≥ 1 are fixed positive integers. We also study those positive integers n, not equal to a prime or twice a prime, such that ϕ(n) = p − 1 holds with some prime p, as well as those positive integers n such that the equation ϕ(n) = f(m) holds with some integer m, where f is a fixed polynomial with integer coefficients and degree degf > 1.     

On Divisible and Torsionfree Modules

A ring R is called left P-coherent in case each principal left ideal of R is finitely presented. A left R-module M (resp. right R-module N) is called D-injective (resp. D-flat) if Ext¹(G, M) = 0 (resp. Tor₁(N, G) = 0) for every divisible left R-module G. It is shown that every left R-module over a left P-coherent ring R has a divisible cover; a left R-module M is D-injective if and only if M is the kernel of a divisible precover A → B with A injective; a finitely presented right R-module L over a left P-coherent ring R is D-flat if and only if L is the cokernel of a torsionfree preenvelope K → F with F flat. We also study the divisible and torsionfree dimensions of modules and rings. As applications, some new characterizations of von Neumann regular rings and PP rings are given.     

Relative T -Injective Modules and Relative T -Flat Modules

Let T be a Wakamatsu tilting module. A module M is called (n, T)-copure injective (resp. (n, T)-copure flat) if ɛ _T ¹ (N, M) = 0 (resp. Γ₁ ^T (N, M) = 0) for any module N with T-injective dimension at most n (see Definition 2.2). In this paper, it is shown that M is (n, T)-copure injective if and only if M is the kernel of an I _n (T)-precover f: A → B with A ∈ Prod T. Also, some results on Prod T-syzygies are presented. For instance, it is shown that every nth Prod T-syzygy of every module, generated by T, is (n, T)-copure injective.     

ON (m,n)-INJECTIVITY OF MODULES

Let R be a ring. For two fixed positive integers m and n, a right R-module M is called (m, n)-injective if every right R-homomorphism from an n-generated submodule of R^m to M extends to one from R^m to M. This definition unifies several definitions on generalizations of injectivity of modules. The aim of this paper is to investigate properties of the (m, n)-injective modules. Various results are developed, many extending known results.     

A Note on Relative Flatness and Coherence

Let R be a ring and M a fixed right R-module. A new characterization of M-flatness is given by certain linear equations. For a left R-module F such that the canonical map M? _R F → Hom _R (M?, F) is injective, where M? = Hom _R (M, R), the M-flatness of F is characterized via certain matrix subgroups. An example is given to show that R need not be M-coherent even if every left R-module is M-flat. Moreover, some properties of M-coherent rings are discussed.     

On Strongly Copure Flat Modules and Copure Flat Dimensions

Let R be a left coherent ring. We first prove that a right R-module M is strongly copure flat if and only if Ext ⁱ (M, C) = 0 for all flat cotorsion right R-modules C and i ≥ 1. Then we define and investigate copure flat dimensions of left coherent rings. Finally, we give some new characterizations of n-FC rings.     

Asymptotic formulas for some retricted partition functions

We use Rademacher’s method to obtain asymptotic expressions for some restricted partition functions. For fixed positive integers m > 1 and l, we consider the number p _m (n) of partitions of n into summands not divisible by m, and the number p _m ^l (n) of partitons of n with the further restriction that any integer occurs at most l times as a summand. 2000 Mathematics Subject Classification Primary—11P82; Secondary—11P83     

Presimplifiable and Cyclic Group Rings

Let R be any commutative ring with identity, and let C be a (finite or infinite) cyclic group. We show that the group ring R(C) is presimplifiable if and only if its augmentation ideal I(C) is presimplifiable. We conjecture that the group rings R(C _n ) are presimplifiable if and only if n = p ^m , p ∈ J(R), p is prime, and R is presimplifiable. We show the necessity of n = p ^m , and we prove the sufficiency when n = 2, 3, 4. These results were made possible by a new formula derived herein for the circulant determinantal coefficients.     

On the power sequence of a graph

Necessary and sufficient conditions for a sequence (p ₁,p ₂, …,p _n ) of positive integers to be the power sequence of a connected graph onn vertices withm edges are given. The maximum power of a connected graph onn vertices withm edges and the class of all extremal graphs are also determined.