We show that each R-module is n-flat (resp., weakly n-flat) if and only if R is an (n,n− 1)-ring (resp., a weakly (n,n− 1)-ring). We also give a new characterization of n-von Neumann regular rings and a characterization of weak n-von Neumann regular rings for (CH)-rings and for local rings. Finally, we show that in a class of principal rings and a class of local Gaussian rings, a weak n-von Neu...

Other than triarylamines, which are protonated by strong acids both at their N-atom and aromatic rings, pyrrole its N-substituted derivatives were exclusively the carbon atoms of heteroaromatic rings. Whereas with trifluoromethane sulfonic acid (TFS) stable pyrrolium salts obtained, weaker trifluoroacetic (TFA) oligomers formed. E.g., from N-(tert-butyl)pyrrole a dimeric compound could be unamb...

let $r$ be an associative ring with identity. an element $x in r$ is called $mathbb{z}g$-regular (resp. strongly $mathbb{z}g$-regular) if there exist $g in g$, $n in mathbb{z}$ and $r in r$ such that $x^{ng}=x^{ng}rx^{ng}$ (resp. $x^{ng}=x^{(n+1)g}$). a ring $r$ is called $mathbb{z}g$-regular (resp. strongly $mathbb{z}g$-regular) if every element of $r$ is $mathbb{z}g$-regular (resp. strongly $...

We develop the theory of “branch algebras”, which are infinitedimensional associative algebras that are isomorphic, up to taking subrings of finite codimension, to a matrix ring over themselves. The main examples come from groups acting on trees. In particular, for every field k we construct a k-algebra K which • is finitely generated and infinite-dimensional, but has only finite-dimensional qu...

The internal strain in cyclic amides are explained as a factor of resonance that are effected on the bond length C-N  and are a major factor of rates of hydrolysis. The cyclic amides in this study are optimized by Gaussian program and the bond length of C-N in the rings are studied by HF/6-31G*.  

The internal strain in cyclic amides are explained as a factor of resonance that are effected on the bond length C-N and are a major factor of rates of hydrolysis. The cyclic amides in this study are optimized by Gaussian program and the bond length of C-N in the rings are studied by HF/6-31G*.

We introduce the notions of τ -exceptional and signed sequences for any finite dimensional algebra. prove that a fixed algebra rank n , positive integer t ≤ there is bijection between set length (basic) ordered support -rigid objects with indecomposable direct summands. If hereditary, our coincide exceptional sequences. The latter were recently introduced by Igusa Todorov, who constructed simil...

In this errata, we reconsider and modify two propositions and their corollaries which were written on epi-retractable and co-epi-retractable modules.

That is, a ring of rank n is a ring (commutative, associative, with unit) whose underlying additive group is isomorphic to Z. The prototypical examples of rings of rank n are, of course, orders in degree n number fields. The class of all such examples consists precisely of those rings of rank n that are integral domains. However, there are many interesting examples of rings of rank n that are n...

Let $R$ be an associative ring with identity. An element $x in R$ is called $mathbb{Z}G$-regular (resp. strongly $mathbb{Z}G$-regular) if there exist $g in G$, $n in mathbb{Z}$ and $r in R$ such that $x^{ng}=x^{ng}rx^{ng}$ (resp. $x^{ng}=x^{(n+1)g}$). A ring $R$ is called $mathbb{Z}G$-regular (resp. strongly $mathbb{Z}G$-regular) if every element of $R$ is $mathbb{Z}G$-regular (resp. strongly $...