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• Basic Exercise Problems in Module Theory Let $R$ be a ring with $1$ and $M$ be a left $R$-module. (a) Prove that $0_Rm=0_M$ for all $m \in M$. Here $0_R$ is the zero element in the ring $R$ and $0_M$ is the zero element in the module $M$, that is, the identity element of the additive group $M$. To simplify the […]
• A Subgroup of Index a Prime $p$ of a Group of Order $p^n$ is Normal Let $G$ be a finite group of order $p^n$, where $p$ is a prime number and $n$ is a positive integer. Suppose that $H$ is a subgroup of $G$ with index $[G:P]=p$. Then prove that $H$ is a normal subgroup of $G$. (Michigan State University, Abstract Algebra Qualifying […]
• If the Localization is Noetherian for All Prime Ideals, Is the Ring Noetherian? Let $R$ be a commutative ring with $1$. Suppose that the localization $R_{\mathfrak{p}}$ is a Noetherian ring for every prime ideal $\mathfrak{p}$ of $R$. Is it true that $A$ is also a Noetherian ring?   Proof. The answer is no. We give a counterexample. Let […]
• If Every Proper Ideal of a Commutative Ring is a Prime Ideal, then It is a Field. Let $R$ be a commutative ring with $1$. Prove that if every proper ideal of $R$ is a prime ideal, then $R$ is a field.   Proof. As the zero ideal $(0)$ of $R$ is a proper ideal, it is a prime ideal by assumption. Hence $R=R/\{0\}$ is an integral […]
• Is the Quotient Ring of an Integral Domain still an Integral Domain? Let $R$ be an integral domain and let $I$ be an ideal of $R$. Is the quotient ring $R/I$ an integral domain?   Definition (Integral Domain). Let $R$ be a commutative ring. An element $a$ in $R$ is called a zero divisor if there exists $b\neq 0$ in $R$ such that […]
• Polynomial Ring with Integer Coefficients and the Prime Ideal $I=\{f(x) \in \Z[x] \mid f(-2)=0\}$ Let $\Z[x]$ be the ring of polynomials with integer coefficients. Prove that \[I=\{f(x)\in \Z[x] \mid f(-2)=0\}\] is a prime ideal of $\Z[x]$. Is $I$ a maximal ideal of $\Z[x]$?   Proof. Define a map $\phi: \Z[x] \to \Z$ defined by \[\phi \left( f(x) […]
• Every Prime Ideal in a PID is Maximal / A Quotient of a PID by a Prime Ideal is a PID (a) Prove that every prime ideal of a Principal Ideal Domain (PID) is a maximal ideal. (b) Prove that a quotient ring of a PID by a prime ideal is a PID.   Proof. (a) Prove that every PID is a maximal ideal. Let $R$ be a Principal Ideal Domain (PID) and let $P$ […]
• Examples of Prime Ideals in Commutative Rings that are Not Maximal Ideals Give an example of a commutative ring $R$ and a prime ideal $I$ of $R$ that is not a maximal ideal of $R$.   Solution. We give several examples. The key facts are: An ideal $I$ of $R$ is prime if and only if $R/I$ is an integral domain. An ideal $I$ of […]