Wenting Tong

Tilting modules of finite projective dimension and a generalization of ∗-modules ✩

Abstract It is well known that tilting modules of projective dimension ⩽ 1 coincide with ∗ -modules generating all injectives. This result is extended in this paper. Namely, we generalize ∗ -modules to so-called ∗ n -modules and show that tilting modules of projective dimension ⩽ n are ∗ n -modules which n-present all injectives.

Multiplication Modules in Which Every Prime Submodule is Contained in a Unique Maximal Submodule

Abstract Let R be a commutative ring. An R-module M is called a multiplication module if for each submodule N of M, N = IM for some ideal I of R. An R-module M is called a pm-module, i.e., M is pm, if every prime submodule of M is contained in a unique maximal submodule of M. In this paper the following results are obtained. (1) If R is pm, then any multiplication R-module M is pm. (2) If M is finitely generated, then M is a multiplication module if and only if Spec(M) is a spectral space if and only if Spec(M) = {PM | P ∈ Spec(R) and P ⊇ M ⊥}. (3) If M is a finitely generated multiplication R-module, then: (i) M is pm if and only if Max(M) is a retract of Spec(M) if and only if Spec(M) is normal if and only if M is a weakly Gelfand module; (ii) M is a Gelfand module if and only if Mod(M) is normal. (4) If M is a multiplication R-module, then Spec(M) is normal if and only if Mod(M) is weakly normal.

A NOTE ON SKEW ARMENDARIZ RINGS

ABSTRACT In this note,we answer a question of Hong et al. (2003) by proving that if α is a monomorphism of a reduced ring R, and R is α-skew Armendariz, then R is α-rigid.

Spectrum of a Noncommutative Ring

R is any ring with identity. Let Spec r (R) (resp. Spec(R)) be the set of all prime right ideals (resp. all prime ideals) of R and let U r (eR) = {P ∈ Spec r (R) | e ∉ P}. In this article, we study the relationships among various ring-theoretic properties and topological conditions on Spec r (R) (with weak Zariski topology). A ring R is called Abelian if all idempotents in R are central (see Goodearl, 1991). A ring R is called 2-primal if every nilpotent element is in the prime radical of R (see Lam, 2001). It will be shown that for an Abelian ring R there is a bijection between the set of all idempotents in R and the clopen (i.e., closed and open) sets in Spec r (R). And the following results are obtained for any ring R: (1) For any clopen set U in Spec r (R), there is an idempotent e in R such that U = U r (eR). (2) If R is an Abelian ring or a 2-primal ring, then, for any idempotent e in R, U r (eR) is a clopen set in Spec r (R). (3) Spec r (R) is connected if and only if Spec(R) is connected.

Comparability, Stability, and Completions of Ideals

Abstract In this paper, we study the properties of 1-comparability and stable range one condition on ideals of regular rings, and we use these results to investigate normalized pseudo-rank functions on ideals and N*-complete ideals. These will generalize the corresponding results of Goodearl. Finally, we give some sufficient conditions under which ℙ(I) is compact.

Spectra of Maximal 1-Sided Ideals and Primitive Ideals

R is any ring with identity. Let Spec r (R) (resp. Max r (R), Prim r (R)) denote the set of all right prime ideals (resp. all maximal right ideals, all right primitive ideals) of R and let U r (eR) = {P ∈ Spec r (R)| e ∉ P}. Let 𝒜 = ∪P∈Prim r (R) Spec r P (R), where Spec r P (R) = {Q ∈Spec r P (R)|P is the largest ideal contained in Q}. A ring is called right quasi-duo if every maximal right ideal is 2-sided. In this article, we study the properties of the weak Zariski topology on 𝒜 and the relationships among various ring-theoretic properties and topological conditions on it. Then the following results are obtained for any ring R: (1) R is right quasi-duo ring if and only if 𝒜 is a space with Zariski topology if and only if, for any Q ∈ 𝒜, Q is irreducible as a right ideal in R. (2) For any clopen (i.e., closed and open) set U in ℬ = Max r (R) ∪ Prim r (R) (resp. 𝒞 = Prim r (R)) there is an element e in R with e 2 − e ∈ J(R) such that U = U r (eR) ∩ ℬ (resp. U = U r (eR) ∩ 𝒞), where J(R) is the Jacobson of R. (3) Max r (R) ∪ Prim r (R) is connected if and only if Max l (R) ∪ Prim l (R) is connected if and only if Prim r (R) is connected.

Generalizations of Semiregular Rings

In this article, we call a ring R right generalized semiregular if for any a ∈ R there exist two left ideals P, L of R such that lr(a) = P⊕ L, where P ⊆ Ra and Ra ∩ L is small in R. The class of generalized semiregular rings contains all semiregular rings and all AP-injective rings. Some properties of these rings are studied and some results about semiregular rings and AP-injective rings are extended. In addition, we call a ring R semi-π-regular if for any a ∈ R there exist a positive integer n and e 2 = e ∈ a n R such that (1 − e)a n ∈ J(R), the Jacobson radical of R. It is shown that a ring R is semi-π-regular if and only if R/J(R) is π-regular and idempotents can be lifted modulo J(R).

ON n- -INJECTIVE ENVELOPES AND n- -PROJECTIVE COVERS

ABSTRACT In this paper the n- -injective (pre)envelopes and the n- -projective (pre)covers are investigated, which generalize and include some known envelopes and covers. Various equivalent characterizations and sufficient conditions of their existence are provided. In addition, the (special) -injective (pre)envelopes are discussed under almost (or quasi)excellent extensions of rings, some new or extending results on the -envelopes and the -covers are obtained.

On Related Power Comparability of Modules

Abstract In this paper, we introduce the concept of generalized cu-rings, study the related power comparability of modules. Let R be an exchange ring, we show that the following are equivalent: (1) For any R-modules A and B, R R ⊕ A ≅ R R ⊕ B implies that there exist a positive integer n and an idempotent e ∈ B(R) such that (2) Given any R-module decompositions M = A 1 ⊕ B 1 = A 2 ⊕ B 2 with A 1 ≅ R R ≅ A 2, there exist n ≥ 1 and C, D, E ≤ M n such that where D is a direct summad of , E is a direct summand of and De = 0, E(1 − e) = 0 for some e ∈ B(R). (3) For any idempotents e, f ∈ R with e = 1 + ab and f = 1 + ba for some a, b ∈ R, there exist u ∈ B(R) and n ≥ 1 such that (4) For any regular element a ∈ R, there exists some n ≥ 1 such that aI n is related unit regular in M n (R). Also, we show that every regular related unit π-regular ring is a generalized cu-ring. These generalize the corresponding results in (1), (2), (3) and (4).

ON ALMOST PRECOVERS AND ALMOST PREENVELOPES

Abstract Let C be a class of R -modules closed under isomorphisms and finite direct sums. It is first shown that the finite direct sum of almost C -precovers is an almost C -precover and the direct sum of an almost C -cover and a weak C -cover is a weak C -cover. Then the notion of almost C -preenvelopes is introduced and studied.