"uniserial on two-sided ideals" idea

[rschwiebert]

By the way, regarding my old "uniserial on two-sided ideals" property suggestion, I found the following in Lam I:

• [ ] (Ex10.5) fully prime <===> all ideals idempotent, and they are linearly ordered by inclusion
• [ ] (Ex10.19) fully semiprime <===> all ideals idempotent
• [ ] (Ex12.16) domain <===> prime AND reversible, reduced <===> semiprime AND reversible

[dyunov]

Y. C. Jeon, N. K. Kim, and Y. Lee, On fully idempotent rings (2010):

The following statements are claimed for the "fully idempotent" = "fully semiprime" property:

• commutative AND domain AND fully semiprime ===> field (Pr1.4)
• fully semiprime ===> center is vN regular (Pr1.5)
• commutative AND fully semiprime ===> vN regular (Pr1.6)
• Morita invariant and passes to corners (Th2.1)
• right duo AND fully semiprime <==> right duo AND strongly regular (Th2.7)
• passes to classical right quotient rings (Th3.1)
• passes to arbitrary direct products (Pr3.2)

Fully semiprime = all ideals are idempotent

Rings are ordered as on here.

• [ ] :x: 112 $\Bbb Q\langle a,b\rangle/(a^2)$. $(b)$.
• [ ] :x: 165 $\Bbb Q\langle x,y\rangle$. $(x)$.
• [ ] :x: 166 $\Bbb Z\langle x_0, \ldots \rangle$. $(x_0)$.
• [ ] :x: 163 Camillo and Nielsen's McCoy ring. $(w)$, where $w = x_{a1} y{a_1}$ is from my comment.
• [ ] :x: 120 Cohn's right-not-left free ideal ring. $(x_0)$.
• [ ] :x: 74 Hurwitz quaternions. $(2)$.
• [ ] :x: 21 Left-not-right Noetherian domain. $(x)$.
• [ ] :x: 75 Lipschitz quaternions. $(2)$.
• [ ] :x: Triangular rings are not even semiprime (the upper right ideal). The unmarked ones are 78, 126, 143.
• [ ] :x: 35 Shepherdson's domain that is not stably finite. $(s)$.

The new condition: ideals linearly ordered by inclusion

• [ ] ideals linearly ordered ===> (sub)directly irreducible. The projection images are ideals and are incomparable.

Every triangular ring has incomparable ideals (the upper row and the lower corner):

• [ ] 87
• [ ] 39
• [ ] 29
• [ ] 78
• [ ] 97
• [ ] 126
• [ ] 20
• [ ] 40
• [ ] 19
• [ ] 30
• [ ] 143
• [ ] 46. Not marked with triangular keyword:
• [ ] 16
• [ ] 13
• [ ] 14.

This section is to be continued.

Remarks

• In $R_{111}$'s description the second sentence looks circular, should it be $F[y]$ instead?
• Is $R{155}$'s maximal ideal really idempotent, as in the notes? [Wikipedia](https://en.wikipedia.org/wiki/Germ(mathematics)#Algebraic_properties) seems to suggest otherwise.
• Also this article might be of interest.

[dyunov]

This paper's Th3 claims that semiprimitive rings are fully semiprime, if simple rings are semiprimitive (Sasiada's paper is a few years 'more recent'). The argument might be salvageable for unital rings, though. I'm writing this comment mostly for myself to read this article later and find out whether it breaks.

Edit. In C.-C. Lai's PhD thesis "Localization in non-Noetherian rings", p32 (A) b), V rings and vN regular rings are said to be fully semiprime. (Even though such rings are mistakenly claimed to be semiprimitive.)

[dyunov]

Related: in "Rings close to regular" (2002) A. Tuganbaev calls a ring right weakly regular if $B^2=B$ for every right ideal $B$.

[dyunov]

H. E. Heatherly, R. P. Tucci, Right weakly regular rings: a survey is about rings whose right ideals are idempotent.