Many arguments in the Theory of Rings and Modules are, on close inspection, purely Lattice theoretic arguments. Cǎlagǎreanu has a long repertoire of such results in his book. The Hopkins-Levitzki Theorem is interesting from this point of view, because a special case of it lends to an obvious lattice theory approach, but the rest is a little more subtle. Albu and Smith have obtained some suffici...

Let $G$ be a countable monoid and let $A$ an Artinian group (resp. module). $\Sigma \subset A^G$ closed subshift which is also subgroup submodule) of $A^G$. Suppose that $\Gamma$ finitely generated consisting pairwise commuting cellular automata \to \Sigma$ are homomorphisms groups modules) with binary operation given by composition maps. We show the valuation action on $\Sigma$ satisfies natur...

A new characteristic property of von Neumann regular rings is proposed in terms of annihilators of elements. An ELT fully idempotent ring is a regular ring whose simple left (or right) modules are either injective or projective. Artinian rings are characterized in terms of Noetherian rings. Strongly regular rings and rings whose two-sided ideals are generated by central idempotents are characte...

Meta-projective modules on a ring R were essentially studied by Feng Lianggui and Tong Wenting in [3] when R is a commutative ring. While Duo-rings were studied by A. L. Fall in [2] and by M. Sanghare in [8]. The results of [3] were extended by M. O. Abdelkader in [5] to the duo-ring case by providing characterizations of Duo-rings. In this note, we give an extension of the MP-dimension notion ...

Let R be a commutative Noetherian ring, a an ideal of R, and let M be a finitely generated R-module. For a non-negative integer t, we prove that H a(M) is a-cofinite whenever H t a(M) is Artinian and H i a(M) is a-cofinite for all i < t. This result, in particular, characterizes the a-cofiniteness property of local cohomology modules of certain regular local rings. Also, we show that for a loca...

Group automata occur in the Krohn-Rhodes Decomposition Theorem and have been extensively investigated in the literature. The incidence rings of group automata were introduced by the first author in analogy with group rings and incidence rings of graphs. The main theorem of the present paper gives a complete description of the structure of incidence rings of group automata in terms of matrix rin...

Suppose R and S are local rings and (R,m) → (S,n) is a flat local homomorphism. Given a finitely generated S-module N , we say N is extended (from R) provided there is an R-module M such that S ⊗R M is isomorphic to N as an S-module. If such a module M exists, it is unique up to isomorphism (cf. [EGA, (2.5.8)], and it is necessarily finitely generated. The m-adic completion R → R̂ and the Hensel...

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...