Typos and minor discrepancies

[dyunov]

Long-term issue as requested by @rschwiebert.

See the edit history for older batches (1, 2, 3: 9 Mar 2024; 4: 10 Apr 2024).

Batch 5 (10 Oct 2024)

• [ ] The name of module 19 is incorrect (as the ring was changed).
• [ ] On the website, in the citation for "An introduction to noncommutative Noetherian rings", the surname Warfield is abbreviated to "W."

[rschwiebert]

Whoa it looks like I misrepresented Norton's ring there. I'll take a look at that. Can you look into the left singular ideal question @dyunov ?

[dyunov]

(regarding the left singular ideal in $R_{78}$)

• The "left" in "maximal left ideal" is redundant. The meaningless side adjective is present in Faith and Page's text.
• It should not be "maximal ideal which is essential in $Q$ " (I don't think there are any maximal ideals in $Q$ that are essential), but rather "maximal among essential ideals"; cf. Chase's example $R_{39}$ and Lam II (2.34) p47 (I think $M$ is the "direct sum").
• $R_{78}$ is isomorphic to the opposite ring of the book's one: So my bet is on that the sides for $P_{73}$ were confused in the note, it should indeed be "Left singular ideal is the subset of strictly upper-triangular matrices" in the description.

[rschwiebert]

@dyunov Thanks for checking that!

I'm not sure I want to make "noetherian complete local" a singular property. I realize it is a bit of an outlier in that chart, but it seemed like a good idea at the time.

[rschwiebert]

Rerphasing R_51 to

Using the field of two elements $F_2$, let $S=F_2[M]$ where $M$ is the monoid of nonnegative real numbers under addition. The required ring is $R=S/I$ where $I$ is the ideal generated by elements of $M$ greater than $1$. (The elements look like linear combinations of elements from the interval $[0,1]$.)

Now the question is whether or not my original representation led to any erroneous properties.

property_id|reason_left|reason_right
91|symmetry with right side|N.C. Norton, Generalizations of the theory of quasi-Frobenius rings p  113
80|symmetry with right side|N.C. Norton, Generalizations of the theory of quasi-Frobenius rings p  113
76|See references to Norton's work|symmetry with left side
22|For any given $r>0$, when $mr > 1$, $x^{mr}=0$. Every proper ideal is contained in such an ideal.|symmetry with left side
10|Every nonunit is nilpotent.|symmetry with left side
45|Every nonunit is nilpotent, so the chain of principal ideals will terminate with $0$ after finitely may steps.|symmetry with left side
90|symmetry with right side|A non-Artinian dual ring such as this one necessarily fails DCC on annihilators.
98|The ring is valuation (uniserial) and Kasch, so it has a simple socle|symmetry with left side
101|The ring is valuation (uniserial) with a nonzero socle, so it's automatically essential.|symmetry with left side
131|Zero dimensional ring trivially is catenary|symmetry with left side
143|since the radical is idempotent, the completion is isomorphic to R/M, not R|symmetry with left side
87|Let $r=1/2$, and consider $ann(r)\to R\to rR$. One can see $ann(r)=((1/2,1])$ is not finitely generated|symmetry with left side

I didn't see anything that becomes invalid: do you? @dyunov

[dyunov]

91 (dual): follows from r(A_i) = B_{1-i} and r(B_i) = A_{1-i}. 80 (not self-injective): written explicitly on p113. 76 (uniserial): not sure about the references, but this follows from the classification of ideals. 22 (nil radical): explanation looks fine 10 (not reduced): obvious 45 (strongly $\pi$-regular): looks fine 90 (DCC annihilator): also follows from the annihilator description. The logic "dual and DCC(ann) [+ commutative?] ===> Artinian" does not seem to be in the DB. 98 (simple socle): again no logic in DB. 101 (essential socle): same. 131 (catenary): the only prime ideal is $J(R)$. 143 (not complete local): I have no expertise here. 87 (not coherent): $ann(r) = B_{1/2}$, it is indeed not f.g.