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A well known problem in the theory of envelopes is to characterize those rings having the property that every (finitely generated, finitely presented, simple) module has a projective (pre)envelope. In this paper, we introduce the $\mathcal{S}$-projective modules for an arbitrary class $\mathcal{S}$ of finitely generated modules, and characterize in this general setting those rings for which every module in $\mathcal{S}$ has a projective preenvelope. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study weak quasi-Armendariz rings which unify the notions of weak Armendariz rings and quasi-Armendariz rings. It is shown that the weak quasi-Armendarizness is a Morita invariant property. For a semiprime ring R, it is shown that R[x]/〈xn〉 is weak quasi-Armendariz, where R[x] is the polynomial ring over R and 〈xn〉 is the ideal of R[x] generated by xn. Various properties of weak quasi-Armendariz rings are also observed. [ABSTRACT FROM AUTHOR]

In this paper we introduce the notion of almost idempotent semirings as the semirings with semilattice additive reduct satisfying the identity x + x2 = x2, and characterize eight subclasses of the variety $\mathcal{AI}$ of all almost idempotent semirings corresponding to the eight subvarieties of the variety $\mathcal{NB}$ of all normal bands. Every almost idempotent semiring S is a distributive lattice of rectangular almost idempotent semirings. Given a semigroup F, the semiring Pf(F) of all finite non-empty subsets of F is almost idempotent precisely when F is a band, and in this case, Pf(F) is freely generated by the band F in the variety $\mathcal{AI}$. This semiring Pf(F) is free in a subclass of $\mathcal{AI}$ if and only if F is in the corresponding subvariety of $\mathcal{NB}$. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring, and let N be a non-zero finitely generated R-module. The purpose of this paper is to show that if I and J are projectively equivalent ideals with respect to N, then a sequence x := x1,...,xn of elements of R is an asymptotic sequence over I with respect to N if and only if it is an asymptotic sequence over J with respect to N. Also, it is shown that if R is local, then the lengths of all maximal asymptotic sequences over an ideal I with respect to N are the same. As a consequence, we derive a generalization of Rees' theorem [23]. [ABSTRACT FROM AUTHOR]

In this paper, we study the class $\widetilde{\mathcal{FI}}$ of complexes of FP-injective left R-modules. It is shown that the pair $(^{\bot}\widetilde{\mathcal{FI}},\widetilde{\mathcal{FI}})$ is a complete cotorsion pair. If R is a left coherent ring, it is proved that every complex has an $\widetilde{\mathcal{FI}}$-cover. We also introduce the FP-injective dimension of complexes. A special attention is paid to the dimension of homologically bounded above R-complexes over a left coherent ring which has a nice functorial description. [ABSTRACT FROM AUTHOR]