Notes from Ring Theory

A list of notes, by date Nov 8 – Densidad…
Nov 15

  • Pere Menal (\cross\sim 1992) Ver si un módulo artiniano tiene \End() semilocal. Rosa Camp & W. Ricki \rightarrow Sí.
  • Krull ’34 \rightarrow Módulos Artinianos satisfacen el Teorema de Krull-Schmidt? No es difícil ver que si {M_R} es artiniano {\Rightarrow} \footnotesize(se forma una cadena descendiente)\normalsize {M_R=N_1\oplus\cdots\oplus N_s}, {N_i} indescomponibles. Es la única descomposición bajo isomorfía y reordenación de los sumandos? (Teorema de Krull-Schmidt). Facchini, H Levy, Vamos ’96 {\rightarrow} No.
  • Un {R}-módulo {M} generado por {r} elementos – {M} es imágen homomórfica de {R^r}

Ene 13

  • Módulos proyectivos finitamente generados {\leadsto} submonoides…
  • “full affine”
  • Monoids are tricky… why? No they’re not. They just have associativity and the presence of an identity
  • Fundamental theorem of abelian groups {\left(\math{N}^k\subseteq\math{Z}^k\right)}
  • {P/PJ(R)} finitely generated {\nRightarrow} {P} finitely generated. Counterexample given by Geramnov, Sakhaev
  • Fair-sized projective modules by Pavel ({A_5})
  • The trace ideal is a bi-lateral ideal: {Tr(P)=\sum_{f\in P^*}{f(P)}}{(P\rightarrow R)\in P^*=\Hom_R(P,R)}. Verify that {\left[Tr(P)\right]^2=Tr(P)}
  • Whitehead, as cited in Pavel’s article: {I} a bi-lateral idempotent ideal of {R} such that {_RI} is finitely generated {\Rightarrow I=Tr(P_R)}, with {P_R} a projective ideal
  • Hyman Bass {\rightarrow} Big proyective ideals

\chapter{Features of the Standard LaTeX Report Class}

1. Section

Use the \verb”

2. Section

” command for major sections, and
the \verb”

2.1. Subsection

” command for subsections, etc.

2.2. Subsection

This is just some text under a subsection.


This is just some text under a subsubsection.


This is just some text under a subsubsubsection.


This is just some text under a subsubsubsubsection.

3. Typesetting Commands

Select a part of the text then click on the button Emphasize (H!),
or Bold (Fs), or Italic (Kt), or Slanted (Kt) to typeset
Emphasize, Bold, Italics,
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You can also typeset \textrm{Roman}, \textsf{Sans Serif},
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You can also apply the special, mathematics only commands
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through Z.

You can apply the size tags — Format menu, Font size submenu —
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large}, {\Large Large}, {\LARGE LARGE}, {\huge huge} and {\Huge

You can use the \verb” etc. ” environment
for typesetting short quotations. Select the text then click on
Insert, Quotations, Short Quotations:

The buck stops here. Harry Truman

Ask not what your country can do for you; ask what you can do for
your country. John F Kennedy

I am not a crook. Richard Nixon

I did not have sexual relations with that woman, Miss Lewinsky.
Bill Clinton

The Quotation environment is used for quotations of more than one
paragraph. Following is the beginning of The Jungle Books
by Rudyard Kipling. (You should select the text first then click
on Insert, Quotations, Quotation):

It was seven o’clock of a very warm evening in the Seeonee Hills
when Father Wolf woke up from his day’s rest, scratched himself,
yawned and spread out his paws one after the other to get rid of
sleepy feeling in their tips. Mother Wolf lay with her big gray
nose dropped across her four tumbling, squealing cubs, and the
moon shone into the mouth of the cave where they all lived.
Augrh” said Father Wolf, “it is time to hunt again.”
And he was going to spring down hill when a little shadow with a
bushy tail crossed the threshold and whined: “Good luck go with
you, O Chief of the Wolves; and good luck and strong white teeth
go with the noble children, that they may never forget the hungry
in this world.”

It was the jackal—Tabaqui the Dish-licker—and the wolves of
India despise Tabaqui because he runs about making mischief, and
telling tales, and eating rags and pieces of leather from the
village rubbish-heaps. But they are afraid of him too, because
Tabaqui, more than any one else in the jungle, is apt to go mad,
and then he forgets that he was afraid of anyone, and runs through
the forest biting everything in his way.

Use the Verbatim environment if you want \LaTeX to preserve
spacing, perhaps when including a fragment from a program such as:

#include // is used for standard libraries.
void main(void) // ”main” method always called first.
cout << ”This is a message.”;
// Send to output stream.

(After selecting the text click on Insert, Code Environments,

4. Mathematics and Text

It holds \cite{KarelRektorys} the following

Theorem 1
(The Currant minimax principle.) Let {T} be completely continuous
selfadjoint operator in a Hilbert space {H}. Let {n} be an
arbitrary integer and let {u_1,\ldots,u_{n-1}} be an arbitrary
system of {n-1} linearly independent elements of {H}. Denote

\displaystyle \max_{\substack{v\in H, v\neq 0\\(v,u_1)=0,\ldots,(v,u_n)=0}}\frac{(Tv,v)}{(v,v)}=m(u_1,\ldots, u_{n-1}) \ \ \ \ \ (1)


Then the {n}-th eigenvalue of {T} is equal to the minimum of these
maxima, when minimizing over all linearly independent systems
{u_1,\ldots u_{n-1}} in {H},

\displaystyle \mu_n = \min_{\substack{u_1,\ldots, u_{n-1}\in H}} m(u_1,\ldots, u_{n-1}) \ \ \ \ \ (2)


The above equations are automatically numbered as equation
(1) and (2).

5. Lists Environments

You can create numbered, bulleted, and description lists (Use the
Itemization or Enumeration buttons, or click on the Insert menu
then chose an item from the Enumeration submenu):

  1. List item 1
  2. List item 2
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    2. Just another list item under a list item.
      1. Third level list item under a list item.
        1. Fourth and final level of list items allowed.
  • Bullet item 1
  • Bullet item 2
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      • Third level bullet item.
        • Fourth (and final) level bullet item.
  • Description List Each description list item has a term
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  • Bunyip Mythical beast of Australian Aboriginal legends.6. Theorem-Like Environments

    The following theorem-like environments (in alphabetical order) are available
    in this style.

    This is an acknowledgement

    This is an algorithm

    This is an axiom

    This is a case

    Claim 1
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    This is a conclusion

    This is a condition

    This is a conjecture

    Corollary 2
    This is a corollary

    This is a criterion

    Definition 3
    This is a definition

    Example 1
    This is an example

    Exercise 1
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    Lemma 4
    This is a lemma

    This is the proof of the lemma.

    This is notation

    This is a problem

    Proposition 5
    This is a proposition

    Remark 1
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    This is a solution

    This is a summary

    Theorem 6
    This is a theorem

    Proof: This is the proof.


    \chapter{The First Appendix}

    The appendix fragment is used only once. Subsequent appendices can be created using the
    Chapter command.

    \chapter{The Second Appendix}

    \chapter{The Third Appendix}

    Some text for the third Appendix.

    This text is a sample for a short bibliography. You can cite a
    book by making use of the command \verb”\cite{KarelRektorys}”:
    \cite{KarelRektorys}. Papers can be cited similarly:
    \cite{Bertoti97}. If you want multiple citations to appear in a
    single set of square brackets you must type all of the citation
    keys inside a single citation, separating each with a comma. Here
    is an example: \cite{Bertoti97, Szeidl2001, Carlson67}.

    \bibitem {KarelRektorys}Rektorys, K., Variational methods in Mathematics,
    Science and Engineering
    , D. Reidel Publishing Company,
    Dordrecht-Hollanf/Boston-U.S.A., 2th edition, 1975

    \bibitem {Bertoti97} \textsc{Bert\'{o}ti, E.}: On mixed variational formulation
    of linear elasticity using nonsymmetric stresses and
    , International Journal for Numerical Methods in
    Engineering., 42, (1997), 561-578.

    \bibitem {Szeidl2001} \textsc{Szeidl, G.}: Boundary integral equations for
    plane problems in terms of stress functions of order one
    , Journal
    of Computational and Applied Mechanics, 2(2), (2001),

    \bibitem {Carlson67} \textsc{Carlson D. E.}: On G\”{u}nther’s stress functions
    for couple stresses
    , Quart. Appl. Math., 25, (1967),


About varasdemate

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