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 module M is called product closed if every hereditary pretorsion class in σ[M ] is closed under products in σ[M ]. Every module which is locally of finite length is product closed and every product closed module is semilocal. LetM ∈ R-Mod be product closed and projective in σ[M ]. It is shown that (1) M is semiartinian; (2) if M is finitely generated then M satisfies the DCC on fully invarian...

We define generalized Koszul modules and rings develop a theory for N-graded with the degree zero part noetherian semiperfect. This specializes to classical graded artinian semisimple developed by Beilinson-Ginzburg-Soergel ungraded semiperfect Green Martinéz-Villa. Let A be left finite ring generated in 1 A0 semiperfect, J its Jacobson radical. By dual of we mean Yoneda Ext Ext_A•(A/J,A/J). If...

In this paper we look at the properties of modules and prime ideals in finite dimensional noetherian rings. This paper is divided into four sections. The first section deals with noetherian one-dimensional rings. Section Two deals with what we define a “zero minimum rings” and explores necessary and sufficient conditions for the property to hold. In Section Three, we come to the minimal prime i...

Band sums of associative rings were introduced by Weissglass in 1973. The main theorem claims that the support of every Artinian band sum of rings is finite. This result is analogous to the well-known theorem on Artinian semigroup rings. 1991 Mathematics subject classification (Amer. Math. Soc): primary 16P20, 16W50; secondary 20M25. Let B be a band, that is, a semigroup consisting of idempoten...

A module $$M$$ is said to be distributive (resp., uniserial) if the submodule lattice of a chain) Any uniserial but ring integers non-uniserial as $$\mathbb{Z}$$ -module. Direct sums (resp. modules are called semidistributive serial) modules. If $$A$$ with automorphism $$\varphi$$ , then we denote by $$A((x,\varphi))$$ skew Laurent series coefficient in which addition naturally defined and mult...

We prove lifting results for DG modules that are akin to Auslander, Ding, and Solberg’s famous lifting results for modules. Introduction Convention. Throughout this paper, let R be a commutative noetherian ring. Hochster famously wrote that “life is really worth living” in a Cohen-Macaulay ring [7]. For instance, if R is Cohen-Macaulay and local with maximal regular sequence t, then R/(t) is ar...

Abstract We show that the image of a subshift X under various injective morphisms symbolic algebraic varieties over monoid universes with variety alphabets is finite type, respectively sofic subshift, if and only so . Similarly, let G be countable A , B Artinian modules ring. prove for every closed submodule $\Sigma \subset A^G$ -equivariant uniformly continuous module homomorphism $\tau \colon...

In a recent paper [17] Miro-Roig, Mezzetti and Ottaviani highlight the link between rational varieties satisfying a Laplace equation and artinian ideals failing the Weak Lefschetz Property. Continuing their work we extend this link to the more general situation of artinian ideals failing the Strong Lefschetz Property. We characterize the failure of the SLP (which includes WLP) by the existence ...

We obtain a complete structural characterization of Cohn-Leavitt algebras over no-exit objects as graded involutive algebras. Corollaries of this result include graph-theoretic conditions characterizing when a Leavitt path algebra is a directed union of (graded) matricial algebras over the underlying field and over the algebra of Laurent polynomials and when the monoid of isomorphism classes of...