Noetherian rings

Let $R$ be a commutative ring with $1$. If $R$ satisfies the a.c.c. (ascending chain condition) for finitely generated ideals then show that $R$ satisfies the a.c.c. for all ideals. Give an example (without proof) of such a ring which is an integral domain but not a p.i.d.

Let $A$ be a Noetherian integral domain integrally closed in its field of fractions $K$. Let $L$ be a finite separable field extension of $K$. If $B$ is the integral closure of $A$ in $L$, outline a proof that $B$ is Noetherian.

Let $F$ be a field, and for $n \geq 1$ let $R_n$ be the subring of the ring of polynomials $F[x]$ consisting of polynomials

such that $f_1 = f_2 = \cdots = f_n = 0$.

Show that the union of a chain $I_{1} \subset I_{2} \subset \cdots$ of ideals of a ring $R$ is an ideal of $R$. (This exercise is referred to in this chapter.)

Show that an integral domain with the property that every strictly decreasing chain of ideals $I_{1} \supset I_{2} \supset \cdots$ must be finite in length is a field.

An ideal $A$ of a commutative ring $R$ with unity is said to be finitely generated if there exist elements $a_{1}, a_{2}, \ldots , a_{n}$ of $A$ such that $A = \left\langle a_{1}, a_{2}, \ldots , a_{n}\right\rangle$. An integral domain $R$ is said to satisfy the ascending chain condition if every strictly increasing chain of ideals $I_{1} \subset I_{2} \subset \cdots$ must be finite in length. Show that an integral domain $R$ satisfies the ascending chain condition if and only if every ideal of $R$ is finitely generated.

Let $R = \mathbb {Z} \oplus \mathbb {Z} \oplus \cdots$ (the collection of all sequences of integers under componentwise addition and multiplication). Show that $R$ has ideals $I_{1}, I_{2}, I_{3}, \ldots$ with the property that $I_{1} \subset I_{2} \subset I_{3} \subset \cdots$. (Thus $R$ does not have the ascending chain condition.)

Prove that a quotient ring of a commutative Noetherian ring is also Noetherian.

Prove that the ring of polynomials in one variable over a commutative Noetherian ring with unity is Noetherian.

Prove that the algebra of polynomials in finitely many variables over a field is Noetherian.