Factorizations of Matrices Over Projective-free Rings

An element of a ring $R$ is called strongly $J^{\#}$-clean provided that it can be written as the sum of an idempotent and an element in $J^{\#}(R)$ that commute. We characterize, in this article, the strongly $J^{\#}$-cleanness of matrices over projective-free rings. These extend many known results on strongly clean matrices over commutative local rings

Developing an Unnatural Amino Acid-Specific Aminoacyl tRNA Synthetase

Unnatural Amino Acids (UAAs), amino acids not present in the human genetic code, have been synthesized to have a broad range of useful properties, in this case, as metal-binders which could have drug delivery applications. In order for the cell to place a UAA into the protein, two components, a unique aminoacyl tRNA synthetase and a corresponding tRNA must be present. If an amino acid is successfully charged to the tRNA, a stop codon is suppressed and a functional protein is built with the UAA at the mutation site. Such a tRNA molecule has previously been developed, as well as many synthetases specific to UAAs. In this work, the range of UAAs which can be incorporated into proteins using the E. coli’s own machinery is expanded by the development of a novel aminoacyl tRNA synthetase. By making a library of synthetase-coding plasmid variants and performing positive and negative screenings, the binding pocket of the synthetase can be modified for specificity to a UAA while not allowing the tRNA to be charged with a natural amino acid. In this work, we are attempting to evolve new tRNA synthetases for the incorporation of metal-binding amino acids by developing the plasmid library and a screening system to find synthetase variants meeting these criteria

Structure theory of central simple ℤd-graded algebras

This paper investigates the structure theory of ℤd- central simple graded algebras and gives the complete decomposition into building block algebras. The results are also applied to generalized Clifford algebras, which are motivating examples of ℤd-central simple graded algebras. © TÜBİTAK

Strongly clean matrices over power series

An n � n matrix A over a commutative ring is strongly clean provided that it can be written as the sum of an idempotent matrix and an invertible matrix that commute. Let R be an arbitrary commutative ring, and let A(x) ∈ Mn (R[[x]]). We prove, in this note, that A(x) ∈ Mn (R[[x]]) is strongly clean if and only if A(0) ∈ Mn(R) is strongly clean. Strongly clean matrices over quotient rings of power series are also determined. � Kyungpook Mathematical Journal

Strongly clean triangular matrix rings with endomorphisms

A ring R is strongly clean provided that every element in R is the sum of an idempotent and a unit that commutate. Let Tn(R; σ) be the skew triangular matrix ring over a local ring R where σ is an endomorphism of R. We show that T2(R; σ) is strongly clean if and only if for any aϵ 1+J(R); b ϵ J(R), la -rσ (b): R→ R is surjective. Further, T3(R; σ) is strongly clean if la-rσ (b); la-rσ2 (b) and lb-rσ (a)are surjective for any a ϵ U(R); b ϵ J(R). The necessary condition for T3(R; σ) to be strongly clean is also obtained. © 2015 Iranian Mathematical Society

Duo property for rings by the quasinilpotent perspective

In this paper, we focus on the duo ring property via quasinilpotent elements, which gives a new kind of generalizations of commutativity. We call this kind of rings qnil-duo. Firstly, some properties of quasinilpotents in a ring are provided. Then the set of quasinilpotents is applied to the duo property of rings, in this perspective, we introduce and study right (resp., left) qnil-duo rings. We show that this concept is not left-right symmetric. Among others, it is proved that if the Hurwitz series ring $H(R; \alpha)$ is right qnil-duo, then $R$ is right qnil-duo. Every right qnil-duo ring is abelian. A right qnil-duo exchange ring has stable range 1

On feckly clean rings

A ring R is feckly clean provided that for any a R there exists an element e R and a full element u R such that a = e + u, eR(1 - e) J(R). We prove that a ring R is feckly clean if and only if for any a R, there exists an element e R such that V(a) V(e), V(1 - a) V(1 - e) and eR(1 - e) J(R), if and only if for any distinct maximal ideals M and N, there exists an element e R such that e M, 1 - e N and eR(1 - e) J(R), if and only if J-spec(R) is strongly zero-dimensional, if and only if Max(R) is strongly zero-dimensional and every prime ideal containing J(R) is contained in a unique maximal ideal. More explicit characterizations are also discussed for commutative feckly clean rings. © 2015 World Scientific Publishing Company

Symmetric modules over their endomorphism rings

Let R be an arbitrary ring with identity and M a right R-module with S=EndR(M). In this paper, we study right R-modules M having the property for f,g∈EndR(M) and for m∈M, the condition fgm=0 implies gfm=0. We prove that some results of symmetric rings can be extended to symmetric modules for this general setting

Symmetric modules over their endomorphism rings

Let R be an arbitrary ring with identity and M a right R-module with S = EndR(M). In this paper, we study right R-modules M having the property for f, g ∈ EndR(M) and for m ∈ M, the condition fgm = 0 implies gfm = 0. We prove that some results of symmetric rings can be extended to symmetric modules for this general setting. © Journal “Algebra and Discrete Mathematics”