In this paper we investigate further properties of fuzzy ideals of a BL-algebra. The notions of fuzzy prime ideals, fuzzy irreducible ideals, and fuzzy Gödel ideals of a BL-algebra are introduced and their several properties are investigated. We give a procedure to generate a fuzzy ideal by a fuzzy set. We prove that every fuzzy irreducible ideal is a fuzzy prime ideal but a fuzzy prime ideal m...

Let  be a local Cohen-Macaulay ring with infinite residue field,  an Cohen - Macaulay module and  an ideal of  Consider  and , respectively, the Rees Algebra and associated graded ring of , and denote by  the analytic spread of  Burch’s inequality says that  and equality holds if  is Cohen-Macaulay. Thus, in that case one can compute the depth of associated graded ring of  as  In this paper we ...

It is proved that a commutative ring is clean if and only if it is Gelfand with a totally disconnected maximal spectrum. It is shown that each indecomposable module over a commutative ring R satisfies a finite condition if and only if R P is an artinian valuation ring for each maximal prime ideal P. Commutative rings for which each indecomposable module has a local endomorphism ring are studied...

A calyx (multiplicative lattice) is a complete lattice endowed with the structure of a monoid such that multiplication by an element is a left adjoint functor of complete lattices (equivalently, a left adjoint functor which preserves colimits). Calyxes are a generalization of the set of ideals of a ring, which form a complete lattice under intersection and summation; in conjunction with the nat...

We introduce the notion of integrality Grothendieck categories as a simultaneous generalization primeness noncommutative noetherian rings and locally schemes. Two different spaces associated to category yield respective definitions integrality, we prove equivalence these using Grothendieck-categorical version Gabriel's correspondence, which originally related indecomposable injective modules pr...

In this article we prove an analogue of Bauer’s theorem from algebraic number theory in the context of hyperbolic systems. 0. Introduction Let φ : X → X be a hyperbolic dynamical system and, for a finite G, let φ̃ : X̃ → X̃ be a G-extension for which φ̃ is also hyperbolic. Given φ we are interested in describing the possibilities for φ̃ in terms of the closed φ-orbits. This is analogous to a classic...

In this paper we study certain families of right ideals in noncommutative rings, called right Oka families, generalizing previous work on commutative rings by T.Y. Lam and the author. As in the commutative case, we prove that the right Oka families in a ring R correspond bijectively to the classes of cyclic right R-modules that are closed under extensions. We define completely prime right ideal...

Nakayama (Ann. of Math. 42, 1941) showed that over an artinian serial ring every module is a direct sum of uniserial modules. Hence artinian serial rings have the property that each right (left) ideal is a finite direct sum of quasi-injective right (left) ideals. A ring with the property that each right (left) ideal is a finite direct sum of quasi-injective right (left) ideals will be called a ...

All rings are commutative with identity and all modules are unital. Let R be a ring, M an R-module and R(M), the idealization of M . Homogenous ideals of R(M) have the form I (+)N , where I is an ideal of R and N a submodule of M such that IM ⊆ N . A ring R (M) is called a homogeneous ring if every ideal of R (M) is homogeneous. In this paper we continue our recent work on the idealization of m...