MATH 3231 Class Notes 01-27-2025

notering-theory

Defined rings.

Example: Group Algebra

$R$ commutative ring. $(G,\cdot )$ finite group, $G=\{e=g_1,g_2,\dots ,g_n\}$ with $x_e,x_{g_2},x_{g_3},\dots \in R[G]$.

“Group algebra” $R[G]=\{r_e{x}_e+r_{g_2}{x}_{g_2}+\dots +r_{g_n}{x}_{g_n}\mid r_{g_i}\in R\}$.

$+$ is “component-wise” → $\left({\sum }_{g\in G}{r}_g{x}_g\right)+\left({\sum }_{g\in G}{r}_g^{\prime }{x}_g\right)={\sum }_{g\in G}(r_g+r_g^{\prime })x_g$

$\cdot$ is induced from

• distributivity
• asking $r\in R$, $r=r\cdot x_e+{\sum }_{g\ne e}0x_g\in R$ to commute with all $R[G]$
• $x_g{x}_h=x_{gh}$

$R[G]$ is a ring with additive identity $0_{R[G]}=0_R$.

If $R$ is unital, $R[G]$ is also unital.

$R[G]$ is commutative if and only if $G$ is commutative.

Example - Ring of Functions

$X$ set, $R$ ring, $R^{\times }=\{f:X\to R\mid f\text{ function}\}$.

$R^{\times }$ is a ring. For $f,g:X\to R$. $f+g\in R^{\times }$ is the function $(f+g)(x)=f(x)+g(x)$ (both in $R$). $+$ is “component-wise”. $f\cdot g$ is similar.

Example - Endomorphism Ring

Defined subrings