Every Maximal Ideal of a Commutative Ring is a Prime Ideal
Problem 351
Let $R$ be a commutative ring with unity.
Then show that every maximal ideal of $R$ is a prime ideal.
Add to solve later Tagged: domain
Add to solve later $(x^3-y^2)$ is a Prime Ideal in the Ring $R[x, y]$, $R$ is an Integral Domain.
Problem 239
Let $R$ be an integral domain. Then prove that the ideal $(x^3-y^2)$ is a prime ideal in the ring $R[x, y]$.
Add to solve later Characteristic of an Integral Domain is 0 or a Prime Number
Problem 228
Let $R$ be a commutative ring with $1$. Show that if $R$ is an integral domain, then the characteristic of $R$ is either $0$ or a prime number $p$.
Add to solve later The Ideal $(x)$ is Prime in the Polynomial Ring $R[x]$ if and only if the Ring $R$ is an Integral Domain
Problem 198
Let $R$ be a commutative ring with $1$. Prove that the principal ideal $(x)$ generated by the element $x$ in the polynomial ring $R[x]$ is a prime ideal if and only if $R$ is an integral domain.
Prove also that the ideal $(x)$ is a maximal ideal if and only if $R$ is a field.
Add to solve later In a Principal Ideal Domain (PID), a Prime Ideal is a Maximal Ideal
Problem 175
Let $R$ be a principal ideal domain (PID) and let $P$ be a nonzero prime ideal in $R$.
Show that $P$ is a maximal ideal in $R$.
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