Talk:Serial module

General corrections

Dear Rschwieb I have just seen what you wrote in Wikipedia about "Serial modules" a month ago. I think you've done a wonderful job, but there is at least a mistake, which should be corrected, and the nicest result of the past 15 years about serial modules are missing, so that the content should be updated. Here are my suggestions.

(1) In "Properties of uniserial and serial rings and modules", you say: "Being right serial is preserved under direct products and preserved under quotients for both rings and modules." This is wrong, because there are direct summands of serial modules that are not serial modules, as was shown by Puninski in "Some model theory over an exceptional uniserial ring and decompositions of serial modules", J. London Math. Soc. 64 (2) (2001), 311-326. I propose that the sentence "Being right serial is preserved under direct products and preserved under quotients for both rings and modules." is changed with "Being right serial is preserved under direct products and preserved under quotients of rings. Being uniserial is preserved for quotients of rings and modules, but never for products. A direct summand of a serial module is not necessarily serial, as was proved by Puninski, but direct summands of {\em finite} direct sums of uniserial modules are serial modules (P\v r\'\i hoda 2004).

(2) In order to update the content, you should mention the monogeny classes and epigeny classes of uniserial modules, in order to present the solution of what Warfield called in his 1975 paper "perhaps the outstanding open problem", that is, the uniqueness question for decompositions of a finitely presented module into uniserial summands (Page 189 of Warfield's paper). I propose to insert in Wikipedia what follows:

Two modules $U$ and $V$ are said to have the same monogeny class, denoted $[U]_m=[V]_m$, if there exist a monomorphism $U\rightarrow V$ and a monomorphism $V\rightarrow U$. Dually, they are said to have the same epigeny class, denoted $[U]_e=[V]_e$, if there exist an epimorphism $U\rightarrow V$ and an epimorphism $V\rightarrow U$.\ The following weak form of the Krull-Schmidt Theorem holds. Let $U_1$, $\dots,$ $U_n$, $V_1$, $\dots,$ $V_t$ be $n+t$ non-zero uniserial right modules over a ring $R$. Then the direct sums $U_1\oplus\dots\oplus U_n$ and $ V_1\oplus\dots\oplus V_t$ are isomorphic $R$-modules if and only if $n=t$ and there exist two permutations $\sigma$ and $\tau$ of $\{1,2,\dots,n\}$ such that $[U_i]_m=[V_{\sigma(i)}]_m$ and $[U_i]_e=[V_{\tau(i)}]_e$ for every $i=1,2,\dots, n$. This result, due to Facchini, has been extended to infinite direct sums of uniserial modules by P\v r\'\i hoda in 2006. This extension involves the so-called quasismall uniserial modules. These modules were defined by Nguyen Viet Dung and Facchini and their existences was proved by Puninski. The weak form of the Krull-Schmidt Theorem holds not only for uniserial modules, but also for several other classes of modules (biuniform modules, cyclically presented modules over serial rings, kernels of morphisms between indecomposable injective modules, couniformly presented modules.)

(3) In the "Textbooks", you should also insert Facchini, Alberto, "Module Theory. Endomorphism rings and direct sum decompositions in some classes of modules", Birkh\"auser Verlag, Basel, 1998.

(4) In the "Primary Sources", you should also insert the following further 5 papers published in the last 15 years:

Facchini, Alberto, Krull-Schmidt fails for serial modules, Trans. Amer. Math. Soc. 348 (1996), 4561–4575.

P\v r\'\i hoda, Pavel, Weak Krull-Schmidt theorem and direct sum decompositions of serial modules of finite Goldie dimension, J. Algebra 281 (2004), 332--341.

P\v r\'\i hoda, Pavel, A version of the weak Krull-Schmidt theorem for infinite direct sums of uniserial modules, Comm. Algebra 34(4) (2006), 1479–1487.

Puninski, Gennadi, Some model theory over a nearly simple uniserial domain and decompositions of serial modules, J. Pure Appl. Algebra,163 (2001), 319–337.

Puninski, Gennadi, Some model theory over an exceptional uniserial ring and decompositions of serial modules, J. London Math. Soc. (2) 64 (2001), no. 2, 311–326.

(5) Maybe, now, you could add at the partial alphabetical list of important contributors to the theory of serial rings A. Facchini, P. P\v r\'\i hoda and G. Puninski as well.

Best regards, Serialsam (talk) 17:51, 24 April 2011 (UTC)

Thank you Serialsam for your feedback on the serial module article: I will be sure to adopt your corrections. I guess while doing these properties for both rings and modules at the same time, I didn't proofread carefully. Glad you pointed them out, and I'll be changing it soon. Rschwieb (talk) 14:53, 25 April 2011 (UTC)

Connection with geometry

A geometer mentioned to me something along the lines of "chain rings have nice geometry". I don't yet know enough about what he's saying to write anything on it, so I'm requesting any geometer who knows what he's talking about to contribute a section called something like "Connection with geometry". Thanks! Rschwieb (talk) 16:33, 26 April 2011 (UTC)