Algebra I - January 2024

ALGEBRA I QUALIFYING EXAM JANUARY 2024

Problem 1.

Let $G$ be a non-trivial group such that $G$ contains a proper subgroup $H$ which contains every proper subgroup of $G$. Show that $G$ is cyclic of order $p^i$ for some prime $p$.

Problem 2.

Let $G$ be a free abelian group of rank $r$. Show that $G$ has only finitely many subgroups of a given finite index $n$.

Problem 3.

1. Suppose that $R$ is a commutative ring with identity. Suppose that $a\in R$ is not nilpotent (meaning there is no integer $n$ such that $a^n=0$). Prove that there is a prime ideal in $R$ that does not contain $a$.

Hint: Consider the set of ideals in $R$ that do not contain any power of $a$.

2. Use the above to show that the set of all nilpotent elements of $R$ is the intersection of all prime ideals of $R$.

Problem 4.

Let $I$ be an ideal of a commutative ring $R$ and $a\in R$. Consider the ideals $I+Ra$ and $(I:a)=\{x\in R\mid ax\in I\}$.

1. Show that $(I+Ra)/Ra\cong I/a(I:a)$.

2. Use the above to show that if $I+Ra$ and $(I:a)$ are both finitely-generated, then $I$ is finitely-generated.

Hint: Five Lemma

Problem 5.

Let $R$ be a PID and $K$ its field of fractions.

1. Show that any subring $L$ of $M_n(K)$ which is finitely-generated as a (unitary) $R$-module, must be free as an $R$-module.

2. A subset $S$ of $K$ has a common denominator in $R$ when there is a nonzero $r\in R$ such that $rS\subset R$.

Let $H$ be a subgroup of $GL_n(K)$ whose matrix entries have a common denominator in $R$.

(i) Consider the subset of $K^n$ given by $M=\sum_{h\in H}h(R^n)\subset K^n$, which consists of finite sums of vectors belonging to some $h(R^n)=\{h(r_1,\ldots,r_n)\mid r_1,\ldots,r_n\in R\}$. Prove that $M\cong R^n$ as an $R$-module.

(ii) Use Part (i) to show that $H$ is conjugate to a subgroup of $GL_n(K)$ with matrix entries in $R$.

Hint: If $e_1,\ldots,e_n$ is the standard basis of $R^n$, then any matrix $A$ in $GL_n(K)$ has $i$-th column given by $A(e_i)\subset K^n$.

Problem 6.

For commutative rings $A,B$ with identity, a ring homomorphism $A\to B$, and non-zero $A$-modules $M,N$ consider the canonical map

given by

1. Find examples of $A,B,A\to B,M$, and $N$ such that the above map is the zero map.

2. Prove that if $M$ is a finitely-generated projective $A$-module, then the above map is an isomorphism.