$C2$ Property of Column Finite Matrix Rings

• Datum: 2014
• Hauptverfasser: Chen, Jianlong, Shen, Liang, Чен, Їанлонг, Шен, Ліанг
• Format: Artikel
• Sprache: Englisch
• Veröffentlicht: Institute of Mathematics, NAS of Ukraine 2014
• Online Zugang: https://umj.imath.kiev.ua/index.php/umj/article/view/2257
• Назва журналу: Ukrains’kyi Matematychnyi Zhurnal

A ring $R$ is called a right $C2$ ring if any right ideal of $R$ isomorphic to a direct summand of $R_R$ is also a direct summand. The ring $R$ is called a right $C3$ ring if any sum of two independent summands of $R$ is also a direct summand. It is well known that a right $C2$ ring must be a right $C3$ ring but the converse assertion is not true. The ring $R$ is called $J$ -regular if $R/J(R)$ is von Neumann regular, where $J(R)$ is the Jacobson radical of $R$. Let $ℕ$ be the set of natural numbers and let $Λ$ be any infinite set. The following assertions are proved to be equivalent for a ring $R$: (1) $ℂFMFM_{ℕ} (R)$ is a right $C2$ ring; (2) $ℂFMFM_{Λ}(R)$ is a right $C2$ ring; (3) $ℂFMFM_{ℕ}(R)$ is a right $C3$ ring; (4) $ℂFMFM_{Λ}(R)$ is a right $C3$ ring; (5) $ℂFMFM_{ℕ}(R)$ is a $J$ -regular ring and $M_n(R)$ is a right $C2$ (or right $C3$) ring for all integers $n ≥ 1$.