### A note on regular rings with stable range one.

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A sequence of Temperley-Lieb algebra elements corresponding to torus braids with growing twisting numbers converges to the Jones-Wenzl projector. We show that a sequence of categorification complexes of these braids also has a limit which may serve as a categorification of the Jones-Wenzl projector.

In the present paper, we will show that the set of minimal elements of a full affine semigroup $A\hookrightarrow {\mathbb{N}}_{0}^{k}$ contains a free basis of the group generated by $A$ in ${\mathbb{Z}}^{k}$. This will be applied to the study of the group ${\text{K}}_{0}\left(R\right)$ for a semilocal ring $R$.

We develop a diagrammatic categorification of the polynomial ring ℤ[x]. Our categorification satisfies a version of Bernstein-Gelfand-Gelfand reciprocity property with the indecomposable projective modules corresponding to xⁿ and standard modules to (x-1)ⁿ in the Grothendieck ring.

In this paper, we introduce related comparability for exchange ideals. Let $I$ be an exchange ideal of a ring $R$. If $I$ satisfies related comparability, then for any regular matrix $A\in {M}_{n}\left(I\right)$, there exist left invertible ${U}_{1},{U}_{2}\in {M}_{n}\left(R\right)$ and right invertible ${V}_{1},{V}_{2}\in {M}_{n}\left(R\right)$ such that ${U}_{1}{V}_{1}A{U}_{2}{V}_{2}=diag({e}_{1},\cdots ,{e}_{n})$ for idempotents ${e}_{1},\cdots ,{e}_{n}\in I$.

Let $R$ be an exchange ring in which all regular elements are one-sided unit-regular. Then every regular element in $R$ is the sum of an idempotent and a one-sided unit. Furthermore, we extend this result to exchange rings satisfying related comparability.

In this paper we investigate the related comparability over exchange rings. It is shown that an exchange ring R satisfies the related comparability if and only if for any regular x C R, there exists a related unit w C R and a group G in R such that wx C G.

We characterize exchange rings having stable range one. An exchange ring $R$ has stable range one if and only if for any regular $a\in R$, there exist an $e\in E\left(R\right)$ and a $u\in U\left(R\right)$ such that $a=e+u$ and $aR\cap eR=0$ if and only if for any regular $a\in R$, there exist $e\in r.ann\left({a}^{+}\right)$ and $u\in U\left(R\right)$ such that $a=e+u$ if and only if for any $a,b\in R$, $R/aR\cong R/bR\u27f9aR\cong bR$.