diff options
Diffstat (limited to 'ga/n1.lyx')
| -rw-r--r-- | ga/n1.lyx | 506 |
1 files changed, 365 insertions, 141 deletions
@@ -349,10 +349,10 @@ Ejemplos: \end_inset son grupos abelianos. -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout La suma es asociativa y conmutativa con elemento neutro 0, y todo elemento \begin_inset Formula $a$ @@ -369,7 +369,11 @@ La suma es asociativa y conmutativa con elemento neutro 0, y todo elemento solo el 0 tiene opuesto. \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate \begin_inset Formula $\mathbb{N}$ \end_inset @@ -391,15 +395,19 @@ La suma es asociativa y conmutativa con elemento neutro 0, y todo elemento \end_inset son monoides conmutativos con el producto. -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout El producto es asociativo y conmutativo con neutro 1, pero el 0 nunca tiene opuesto. \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate Llamamos \begin_inset Formula $Y^{X}$ @@ -423,10 +431,10 @@ Llamamos \end_inset es un monoide, pero no es conmutativo si hay al menos dos elementos. -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout Claramente \begin_inset Formula $\circ$ \end_inset @@ -471,7 +479,11 @@ Claramente . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate Llamamos \series bold @@ -499,14 +511,18 @@ grupo simétrico \end_inset es un grupo. -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout Es asociativa, tiene como neutro la identidad y todo elemento es invertible. \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate Sea \begin_inset Formula $X$ @@ -530,10 +546,10 @@ Sea es un monoide conmutativo cuyos elementos invertibles son las funciones que no se anulan. -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout Ambas operaciones son conmutativas y asociativas, la suma tiene como neutro la función constante 0 y el producto la función constante 1. El inverso de una función @@ -563,7 +579,11 @@ Ambas operaciones son conmutativas y asociativas, la suma tiene como neutro no se anula. \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Standard Dada una operación \begin_inset Formula $*$ @@ -605,17 +625,21 @@ Si \end_inset tiene a lo sumo un neutro. -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout \begin_inset Formula $f=e*f=e$ \end_inset . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Standard Dado un monoide \begin_inset Formula $(X,*)$ @@ -655,17 +679,21 @@ Si \end_inset tiene a lo sumo un simétrico. -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout \begin_inset Formula $y=e*y=(x*a)*y=x*(a*y)=x*e=x$ \end_inset . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate Si \begin_inset Formula $a$ @@ -673,10 +701,10 @@ Si tiene simétrico por un lado, es cancelable por dicho lado. En particular, todo elemento invertible es cancelable. -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout Si, por ejemplo, \begin_inset Formula $a$ \end_inset @@ -696,7 +724,11 @@ Si, por ejemplo, . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Section Anillos \end_layout @@ -979,10 +1011,10 @@ Todo elemento invertible es regular. . En particular, el 0 y el 1 son únicos. -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout \begin_inset Formula $b+a=a\implies b=b+(a-a)=(b+a)-a=a-a=0$ \end_inset @@ -993,7 +1025,11 @@ Todo elemento invertible es regular. . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate El opuesto de \begin_inset Formula $a$ @@ -1011,10 +1047,10 @@ El opuesto de \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout \begin_inset Formula $0a+0a=(0+0)a=0a=0a+0\implies0a=0$ \end_inset @@ -1025,16 +1061,20 @@ El opuesto de se prueba análogamente. \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate \begin_inset Formula $a(-b)=(-a)b=-(ab)$ \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout \begin_inset Formula $a(-b)+ab=a(-b+b)=a0=0$ \end_inset @@ -1058,23 +1098,31 @@ El opuesto de se prueba análogamente. \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate \begin_inset Formula $a(b-c)=ab-ac$ \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout \begin_inset Formula $a(b-c)=a(b+(-c))=ab+a(-c)=ab+(-ac)=ab-ac$ \end_inset . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate \begin_inset Formula $a$ \end_inset @@ -1096,9 +1144,9 @@ El opuesto de \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper \begin_layout Enumerate \begin_inset Argument item:1 status open @@ -1119,6 +1167,7 @@ Basta ver que . \end_layout +\begin_deeper \begin_layout Enumerate \begin_inset Argument item:1 status open @@ -1169,6 +1218,11 @@ Tenemos \end_layout \end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate Si \begin_inset Formula $0=1$ @@ -1179,17 +1233,21 @@ Si \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout \begin_inset Formula $a\in A\implies a=a1=a0=0$ \end_inset . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Standard Dado un anillo \begin_inset Formula $A$ @@ -1278,10 +1336,10 @@ Propiedades: Dados un anillo \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout Para \begin_inset Formula $n=0$ \end_inset @@ -1315,16 +1373,20 @@ Para . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate \begin_inset Formula $(n+m)a=na+ma$ \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout Para \begin_inset Formula $m=0$ \end_inset @@ -1358,16 +1420,20 @@ Para . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate \begin_inset Formula $n(ma)=(nm)a$ \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout Para \begin_inset Formula $n=0$ \end_inset @@ -1392,7 +1458,11 @@ Para . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate Si \begin_inset Formula $n,m\geq0$ @@ -1415,10 +1485,10 @@ Si \end_inset enteros arbitrarios. -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout Para \begin_inset Formula $m=0$ \end_inset @@ -1453,7 +1523,7 @@ Para \end_layout -\begin_layout Standard +\begin_layout Plain Layout Primero vemos que, para \begin_inset Formula $m>0$ \end_inset @@ -1502,7 +1572,7 @@ Primero vemos que, para . \end_layout -\begin_layout Standard +\begin_layout Plain Layout Con esto, sea \begin_inset Formula $m>0$ \end_inset @@ -1518,7 +1588,11 @@ Con esto, sea . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate Si \begin_inset Formula $A$ @@ -1545,10 +1619,10 @@ Si \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout Para \begin_inset Formula $n=0$ \end_inset @@ -1586,7 +1660,11 @@ Para . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Section Subanillos \end_layout @@ -1769,7 +1847,8 @@ Para que \end_inset . -\end_layout +\begin_inset Note Comment +status open \begin_layout Description \begin_inset Formula $[1\implies2]$ @@ -1862,6 +1941,11 @@ Para que , luego es cerrado para sumas. \end_layout +\end_inset + + +\end_layout + \begin_layout Standard Algunos subanillos: \end_layout @@ -1915,9 +1999,9 @@ Cada uno de \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper \begin_layout Enumerate \begin_inset Argument item:1 status open @@ -1950,6 +2034,7 @@ Si . \end_layout +\begin_deeper \begin_layout Enumerate \begin_inset Argument item:1 status open @@ -1967,6 +2052,11 @@ Obvio. \end_layout \end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate Llamamos \series bold @@ -1985,10 +2075,10 @@ subanillo primo \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout Claramente \begin_inset Formula $\mathbb{Z}1$ \end_inset @@ -2039,7 +2129,11 @@ Claramente . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate Si \begin_inset Formula $A$ @@ -2062,14 +2156,18 @@ Si \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout No contiene al 1. \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate Dado \begin_inset Formula $z\in\mathbb{C}$ @@ -2277,49 +2375,61 @@ Propiedades: Sean \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout \begin_inset Formula $0+f(0)=f(0)=f(0+0)=f(0)+f(0)\implies0=f(0)$ \end_inset . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate \begin_inset Formula $f(-a)=-f(a)$ \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout \begin_inset Formula $f(a)+f(-a)=f(a+(-a))=f(0)=0$ \end_inset . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate \begin_inset Formula $f(a-b)=f(a)-f(b)$ \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout \begin_inset Formula $f(a-b)=f(a)+f(-b)=f(a)-f(b)$ \end_inset . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate \begin_inset Formula $f(a_{1}+\dots+a_{n})=f(a_{1})+\dots+f(a_{n})$ \end_inset @@ -2332,10 +2442,10 @@ Propiedades: Sean \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout Para \begin_inset Formula $n=0$ \end_inset @@ -2347,7 +2457,11 @@ Para . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate Si \begin_inset Formula $a$ @@ -2362,10 +2476,10 @@ Si \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout \begin_inset Formula $f(a)f(a^{-1})=f(aa^{-1})=f(1)=1$ \end_inset @@ -2376,7 +2490,11 @@ Si . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate \begin_inset Formula $f(a_{1}\cdots a_{n})=f(a_{1})\cdots f(a_{n})$ \end_inset @@ -2402,10 +2520,10 @@ Si \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout \begin_inset Formula $1=f(1)\in f(A')$ \end_inset @@ -2437,7 +2555,11 @@ Si . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate Si \begin_inset Formula $B'$ @@ -2456,10 +2578,10 @@ Si \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout \begin_inset Formula $1\in f^{-1}(1)\in f^{-1}(B')$ \end_inset @@ -2491,7 +2613,11 @@ Si . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate Si \begin_inset Formula $f$ @@ -2502,10 +2628,10 @@ Si \end_inset también. -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout \begin_inset Formula $f^{-1}(1)=1$ \end_inset @@ -2545,7 +2671,11 @@ Si . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Standard Ejemplos: \end_layout @@ -2572,9 +2702,9 @@ Dados anillos \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper \begin_layout Enumerate \begin_inset Argument item:1 status open @@ -2595,6 +2725,7 @@ status open . \end_layout +\begin_deeper \begin_layout Enumerate \begin_inset Argument item:1 status open @@ -2624,6 +2755,11 @@ status open \end_layout \end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate Sea \begin_inset Formula $B$ @@ -2662,10 +2798,10 @@ Dado un anillo \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout \begin_inset Formula $\mu(1)=1$ \end_inset @@ -2701,7 +2837,11 @@ Dado un anillo . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate Dada una familia de anillos \begin_inset Formula $(A_{i})_{i\in I}$ @@ -2810,7 +2950,12 @@ ideal \end_inset . - Todo ideal contiene al 0, pues tomando + Todo ideal contiene al 0 +\begin_inset Note Comment +status open + +\begin_layout Plain Layout +, pues tomando \begin_inset Formula $a\in I$ \end_inset @@ -2818,6 +2963,11 @@ ideal \begin_inset Formula $0=a+(-1)a\in I$ \end_inset + +\end_layout + +\end_inset + . \end_layout @@ -2905,10 +3055,10 @@ ideal principal \end_inset son de esta forma. -\end_layout +\begin_inset Note Comment +status open -\begin_deeper -\begin_layout Standard +\begin_layout Plain Layout Sea \begin_inset Formula $I$ \end_inset @@ -2984,7 +3134,11 @@ Sea . \end_layout -\end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate Sean \begin_inset Formula $I$ @@ -3092,7 +3246,11 @@ anillo cociente de \series default . - +\begin_inset Note Comment +status open + +\begin_layout Plain Layout + \series bold Demostración: \series default @@ -3173,6 +3331,11 @@ Demostración: . \end_layout +\end_inset + + +\end_layout + \begin_layout Standard Es claro que \begin_inset Formula $A/0\cong A$ @@ -3192,7 +3355,11 @@ Es claro que \end_inset . - En efecto, dado +\begin_inset Note Comment +status open + +\begin_layout Plain Layout +En efecto, dado \begin_inset Formula $a\in\mathbb{Z}$ \end_inset @@ -3217,12 +3384,17 @@ Es claro que \end_inset , -\begin_inset Formula $a\equiv b\iff a-b\in n\mathbb{Z}\iff n|a-b\overset{|a-b|<n}{\iff}a=b$ +\begin_inset Formula $a\equiv b\iff a-b\in n\mathbb{Z}\iff n\mid a-b\overset{|a-b|<n}{\iff}a=b$ \end_inset . \end_layout +\end_inset + + +\end_layout + \begin_layout Standard Dado un anillo conmutativo \begin_inset Formula $A$ @@ -3240,9 +3412,9 @@ Dado un anillo conmutativo \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper \begin_layout Enumerate \begin_inset Argument item:1 status open @@ -3267,6 +3439,7 @@ Dado . \end_layout +\begin_deeper \begin_layout Enumerate \begin_inset Argument item:1 status open @@ -3296,6 +3469,11 @@ En particular \end_layout \end_deeper +\end_inset + + +\end_layout + \begin_layout Enumerate Un ideal \begin_inset Formula $I$ @@ -3318,9 +3496,9 @@ Un ideal \end_inset . -\end_layout +\begin_inset Note Comment +status open -\begin_deeper \begin_layout Description \begin_inset Formula $[1\implies2\implies3]$ \end_inset @@ -3328,6 +3506,7 @@ Un ideal Obvio. \end_layout +\begin_deeper \begin_layout Description \begin_inset Formula $[3\implies1]$ \end_inset @@ -3356,6 +3535,11 @@ Un ideal \end_layout \end_deeper +\end_inset + + +\end_layout + \begin_layout Standard Sea \begin_inset Formula $f:A\to B$ @@ -3391,7 +3575,11 @@ núcleo \end_inset . - +\begin_inset Note Comment +status open + +\begin_layout Plain Layout + \series bold Demostración: \series default @@ -3472,6 +3660,11 @@ Demostración: . \end_layout +\end_inset + + +\end_layout + \begin_layout Standard Un homomorfismo de anillos \begin_inset Formula $f:A\to B$ @@ -3482,7 +3675,8 @@ Un homomorfismo de anillos \end_inset . -\end_layout +\begin_inset Note Comment +status open \begin_layout Itemize \begin_inset Argument item:1 @@ -3548,6 +3742,11 @@ Sean es inyectiva. \end_layout +\end_inset + + +\end_layout + \begin_layout Standard \series bold @@ -3947,7 +4146,11 @@ Sean \end_inset . - En efecto, +\begin_inset Note Comment +status open + +\begin_layout Plain Layout +En efecto, \begin_inset Formula $(n)(m)=(\{ab\}_{a\in(n),b\in(m)})=(\{pnqm\}_{p,q\in\mathbb{Z}})=(\{knm\})_{k\in\mathbb{Z}}=(nm)$ \end_inset @@ -3962,6 +4165,11 @@ Sean . \end_layout +\end_inset + + +\end_layout + \begin_layout Section Teoremas de isomorfía \end_layout @@ -4110,6 +4318,11 @@ Así, si \begin_inset Formula $\frac{A\times B}{0\times B}\cong A$ \end_inset + +\begin_inset Note Comment +status open + +\begin_layout Plain Layout , pues el homomorfismo de proyección \begin_inset Formula $f:A\times B\to A$ \end_inset @@ -4122,6 +4335,11 @@ Así, si \begin_inset Formula $0\times B$ \end_inset + +\end_layout + +\end_inset + . \end_layout @@ -4421,7 +4639,8 @@ característica \end_inset . -\end_layout +\begin_inset Note Comment +status open \begin_layout Description \begin_inset Formula $[1\implies2]$ @@ -4592,6 +4811,11 @@ característica . \end_layout +\end_inset + + +\end_layout + \begin_layout Standard \series bold |