Let $(𝔄,{ϵ}_u)$ and $(𝔅,{ϵ}_b)$ be two pointed sets. Given a family of three maps $ℱ=\{f_1:𝔄\to 𝔄;f_2:𝔄\times 𝔄\to 𝔄;f_3:𝔄\times 𝔄\to 𝔅\}$, this family provides an adequate decomposition of $𝔄\setminus \left\{{ϵ}_u\right\}$ as the orthogonal disjoint union of well-described $ℱ$-invariant subsets. This decomposition is applied to the structure theory of graded involutive algebras, graded quadratic algebras and graded weak $H^{*}$-algebras.

Let $D$ be a division ring finite dimensional over its center $F$. The goal of this paper is to prove that for any positive integer $n$ there exists $a\in D^{\left(n\right)},$ the $n$th multiplicative derived subgroup such that $F\left(a\right)$ is a maximal subfield of $D$. We also show that a single depth-$n$ iterated additive commutator would generate a maximal subfield of $D.$

We study certain subgroups of the Hopf group-coalgebra automorphism group of Radford’s $\pi$-biproduct. Firstly, we discuss the endomorphism monoid ${\mathrm{End}}_{\pi \text{-Hopf}}(A\times H,p)$ and the automorphism group ${\mathrm{Aut}}_{\pi \text{-Hopf}}(A\times H,p)$ of Radford’s $\pi$-biproduct $A\times H={\{A\times H_{\alpha }\}}_{\alpha \in \pi }$, and prove that the automorphism has a factorization closely related to the factors $A$ and $H={\left\{H_{\alpha }\right\}}_{\alpha \in \pi }$. What’s more interesting is that a pair of maps $(F_L,F_R)$ can be used to describe a family of mappings $F={\left\{F_{\alpha }\right\}}_{\alpha \in \pi }$. Secondly, we consider the relationship between the automorphism group ${\mathrm{Aut}}_{\pi \text{-Hopf}}(A\times H,p)$ and the automorphism group ${\mathrm{Aut}}_{\pi \text{-}𝒴𝒟\text{-Hopf}}\left(A\right)$ of $A$, and...

Let $R$ be an associative ring and $M$ be a left $R$-module. We introduce the concept of the incidence module $I(X,M)$ of a locally finite partially ordered set $X$ over $M$. We study the properties of $I(X,M)$ and give the necessary and sufficient conditions for the incidence module to be an IN-module, -module, nil injective module and nonsingular module, respectively. Furthermore, we show that the class of -modules is closed under direct product and upper triangular matrix modules.

We give some sufficient and necessary conditions for an element in a ring to be an EP element, partial isometry, normal EP element and strongly EP element by using solutions of certain equations.