## 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
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2.1. Subsection

” command for subsections, etc.

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

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• Bunyip Mythical beast of Australian Aboriginal legends.6. Theorem-Like Environments

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\appendix

\chapter{The First Appendix}

The appendix fragment is used only once. Subsequent appendices can be created using the
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This text is a sample for a short bibliography. You can cite a
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{9}
\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
displacements
, 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),
237-261.

\bibitem {Carlson67} \textsc{Carlson D. E.}: On G\”{u}nther’s stress functions
for couple stresses
, Quart. Appl. Math., 25, (1967),
139-146.

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