Generalizations of von Neumann regularity, I

[dyunov]

1) from Huh, Kim, Lee, "Examples of strongly $\pi$-regular rings":

• [ ] Cor3.4 p206: semilocal + nil radical + right Rickart => strongly $\pi$-regular.
• [ ] Cor3.6 p206: right continuous + nil radical + 2-primal => strongly $\pi$-regular. 2) from Y. Hirano, "Some studies on strongly $\pi$-regular rings": [note that "N-ring" means "2-primal" (quotient by prime radical $P(R)$ is reduced, or equivalently, $P(R) = N(R)$ ). The equivalence is stated by Birkenmeier, Kim, Park "Regularity conditions and the simplicity of prime factor rings" p215 (attributed to Shin).]
• [ ] Th1: for 2-primal rings, TFAE: 1) strongly $\pi$-regular, 2) $\pi$-regular, 4) $J(R)$ is nil and $R/J(R)$ is $\pi$-regular, 6) $R/P(R)$ is strongly regular, 9) matrix rings over $R$ are strongly $\pi$-regular.
• [ ] Th2: for Abelian rings, TFAE: 1) strongly $\pi$-regular, 2) $\pi$-regular, 3) $J(R)$ is nil and $R/J(R)$ is $\pi$-regular, 5) $R/P(R)$ is $\pi$-regular, 8) $I_0$ + some annihilator condition.
• [ ] Th4: for Abelian rings, TFAE: 1) strongly $\pi$-regular, 2) $\pi$-regular, 5) matrix rings over $R$ are strongly $\pi$-regular. 3) from Rowen, "Examples of semiperfect rings":
• [ ] Ex2.4. A semiperfect ring with nil radical that is not strongly $\pi$-regular. The construction is difficult and uses the Golod-Shafarevich theorem. Rowen writes "not left $\pi$-regular", Dischinger's result was apparently unknown to him.

[rschwiebert]

@dyunov What's Dischinger's result you're referring to? (maybe you have it in a separate issue but we could use the reference here.) Is it a result that establishes symmetr of the π-regular condition?

[dyunov]

@rschwiebert Yes, exactly that (under the definition "chains of form $Ra \supset Ra^2 \supset Ra^3 \supset \ldots$ stabilize").

Edit. There's a confusion in terminology, some authors call it strongly $\pi$-regular and others just $\pi$-regular like Rowen. In DaRT it's strongly $\pi$-regular.