Representations of *-regular rings and their ortholattices of projections

On linear representation of $\ast$-regular rings having representable ortholattice of projections

This paper aims at the following results: \begin{enumerate} \item The class of all $*$-regular rings forms a variety. \item A subdirectly irreducible $*$-regular ring $R$ is faithfully representable (i.e. isomorphic to a subring of an endomorphisms ring of vector spaces, where the involution is given by adjunction with respect to a scalar product on the vector space) if so is its ortholattice of projections. \end{enumerate} This is a short version of the second author's PhD thesis.

带对合的可表示正则环的直接有限性：一个反例

Direct finiteness of representable regular rings with involution: A counterexample

受 Bruns–Roddy 的不可嵌入模正交格启发，本文借助 ℓ² 等内积空间上的移位算子构造 *-正则 *-环，给出可表示正则环直接有限性的反例。

A Note on $*$-Clean Rings

A $*$-ring $R$ is called (strongly) $*$-clean if every element of $R$ is the sum of a projection and a unit (which commute with each other). In this note, some properties of $*$-clean rings are considered. In particular, a new class of $*$-clean rings which called strongly $π$-$*$-regular are introduced. It is shown that $R$ is strongly $π$-$*$-regular if and only if $R$ is $π$-regular and every idempotent of $R$ is a projection if and only if $R/J(R)$ is strongly regular with $J(R)$ nil, and every idempotent of $R/J(R)$ is lifted to a central projection of $R.$ In addition, the stable range c…

投影代数与自由投影生成及幂等元生成的正则*-半群

Projection algebras and free projection- and idempotent-generated regular $*$-semigroups

本文引入自由投影生成的正则*-半群 $PG(P)$，证明投影代数范畴在正则*-半群范畴中余反射，并建立投影代数与正则*-双序集的范畴等价，同时给出其与自由幂等元生成半群、拓扑广群及 Temperley-Lieb 幺半群的联系。

具有正规投影的范畴中的阿贝尔对象

Abelian objects in categories with normal projections

本文证明在具有正规投影的任意范畴中，内部阿贝尔群仍保持简单行为，即与减法代数等价、结构唯一且态射保持结构，将幺半与减法范畴的结论推广。