MATH 3231 Class Notes 02-07-2025

notering-theory

Prime Ideals

Recall

$P⊴R$ is prime if $xy\in P⟹x$ or $y\in P$ (and $P\ne R$).

Remark

$f:R\to S$ morphism and $q⊴S$ prime ideal, then $f^{-1}q⊴R$ is a prime ideal.

Proof

$x,y\in R$, and $x\cdot y\in f^{-1}q⟹f(x)f(y)=f(xy)\in q$.

Remark

If $I⊴R$ and $I\subseteq P⊴R$, with $P$ a prime ideal, then ${}^P{/}_I⊴{}^R{/}_I$ is prime.

Proof

If $x+I,y+I\in {}^R{/}_I$ (where $x,y\in R$), $(x+I)(y+I)=(xy+I)\in {}^P{/}_I$, then $xy+I=z+I$ for some $z\in P$, so $xy=\underset{\in P}{z}+\underset{\in I\subseteq P}{t},t\in I$. Hence $x$ or $y$ are in $P$, so $x+I$ or $y+I$ are in $P$.

Corollary: Correspondence Theorem for Quotients

Defined Domain

Remark

If $u\in R$ is a unit ($u$ has a multiplicative inverse in $R$), then $u$ is not a zero-divisor.

Proof

$\phi :R\to R$ defined by $\phi (x)=ux$. $u$ is a unit $⟺$ $\phi$ is bijective $⟹$ $\phi$ injective $⟺$ $u$ is not a zero-divisor.

Defined Field