Prime and Maximal Ideals

Let \(A := \mathbb {R}[x,y]/(x^2+y^2+1)\). Let \(M \subset A\) be a maximal ideal. Determine the field \(A/M\).

Find all the maximal ideals in the following rings:

1. \[ \mathbb {Q}[x,y]/\langle xy-2,x^2-y^2\rangle \]
2. \[ \mathbb {R}[x,y]/\langle xy-2,x^2-y^2\rangle \]
3. \[ \mathbb {C}[x,y]/\langle xy-2,x^2-y^2\rangle \]

Consider the maximal ideal \(\langle x,y\rangle \subset k[x,y]\) for a field \(k\).

1. Prove that the kernel of the natural map of \(k[x, y]\)-modules:

   \[ \langle x,y\rangle \otimes _{k[x,y]}\langle x,y\rangle \to \langle x,y\rangle ^2 \]

   is a one-dimensional vector space over \(k\) and find a generator of it.

2. Find the other two Tors:

   \[ \operatorname {Tor}^{k[x,y]}_1(\langle x,y\rangle ,\langle x,y\rangle )\quad \text {and}\quad \operatorname {Tor}^{k[x,y]}_2(\langle x,y\rangle ,\langle x,y\rangle ) \]

Let \(k\) be the field with 2 elements. Identify all the maximal ideals in the ring \(R=k[x,y]/(xy,x^2+y+1)\).

Let \(k=\mathbb {Z}/3\mathbb {Z}\) be the field with three elements. Find all the prime ideals in the ring \(R=k[x,y]/(x^3-x,y^2+x)\).

Find every prime ideal \(Q\) of \(R=\mathbb {Z}[x]/(x^3-3x,4+x)\) and count the number of elements in each quotient ring

\(R/Q\).

Let \(R\) be a commutative ring with identity such that \(IJ = I \cap J\) for all ideals \(I\) and \(J\). Prove that each prime ideal of \(R\) is maximal.

State the Noether normalization lemma. Let \(M\) be a maximal ideal of the ring \(A=\mathbb {R}[x,y,z]/(x^2+y^2+z^2+1)\). Determine the field \(A/M\).

Let \(R\) be the ring of continuous real-valued functions on the closed interval \([0,1]\). What are the maximal ideals of \(R\)?

Determine all maximal ideals of the ring \(\mathbb {C}[x,y]/(x^3-x^2y,xy^2+xy+x+1)\).

Describe all the prime ideals in the ring \(\mathbb {Z}[x]/(x^3+1,6)\).

Identify all the prime ideals in the ring \((\mathbb {Z}/14)[x,y]/(2y-1,x^2+x-y^2)\).

Consider the ideal \(I=(2,1+\sqrt {-5})\) in the ring \(R=\mathbb {Z}[\sqrt {-5}]\). Is \(I\) a prime ideal? Is \(I\) a projective \(R\)-module?

Find all the prime ideals of \(\mathbb {Z}[x]/(30,x^3+1)\) and identify which are maximal.

Find all prime ideals of the ring \(\mathbb {Z}[x]/(15,x^3-2)\).

Determine all prime ideals of the ring \(\mathbb {R}[x,ix]\), where \(x\) is an indeterminate over \(\mathbb {R}\), and \(i^2=-1\).

Find all the maximal ideals of \(\mathbb {Z}[\sqrt {-13}]\) that contain 7, and prove that their product is precisely the ideal \((7)\).

Let \(R=\mathbb {Z}[x]\), set \(I=(x^3)\), and let \(U\) be the multiplicatively closed set \(\{n\in \mathbb {Z}\mid n\text { is odd}\}\). Find all the prime ideals in the ring \((U^{-1}R)/U^{-1}I\).

1. List the prime ideals of the ring \(R=\mathbb {Z}[x,y]/\langle 3+x, y(y-1)+x^2, x\rangle \).
2. Give an example of an integral domain with exactly 2 prime ideals.

Let \(R\) be a commutative ring with 1.

1. If a maximal ideal \(\mathfrak {m}\) of \(R\) is principal, prove that there is no ideal \(I\) with \(\mathfrak {m}^2\subsetneq I\subsetneq \mathfrak {m}\).
2. Give an example where \(\mathfrak {m}\) is maximal, but there exists an ideal \(I\) with \(\mathfrak {m}^2\subsetneq I\subsetneq \mathfrak {m}\).

Let \(R\) be a commutative ring with \(1\ne 0\). Recall that the nilradical of \(R\) is the ideal \[ \mathcal {N}(R)=\{x\in R:x^n=0\text { for some positive integer }n\}. \] Prove that the following are equivalent:

1. \(R\) has exactly one prime ideal;
2. \(R/\mathcal {N}(R)\) is a field.

Let \(R\) be a ring with 1. A left \(R\)-module \(M\) is called simple if \(M\ne 0\) and if the only submodules of \(M\) are \(M\) and 0. Show that every simple module is isomorphic to \(R/I\) for some maximal left ideal \(I\) and that \(I\) is unique if \(R\) is commutative.

Let \(R\) be a commutative ring, and let \(S\) be a multiplicative subset of \(R\) (\(1\in S\), and if \(s\) and \(t\) are in \(S\), then \(st\) is in \(S\)). Show that an ideal \(I\) that is maximal with the property that \(S\cap I=\emptyset \) is prime.

Let \(R\) be a commutative ring. If for each prime ideal \(\mathfrak {p}\) of \(R\), the local ring \(R_{\mathfrak {p}}\) contains no nonzero nilpotent elements, prove that \(R\) contains no nonzero nilpotent elements.

Let \(\mathfrak {a}\) be the ideal of the polynomial ring \(\mathbb {R}[x]\) that is generated by \(x^3-5x^2+6x-2\) and \(x^4-4x^3+3x^2-4x+2\). Is \(\mathfrak {a}\) a principal ideal? Is it prime? Is it maximal?

Let \(m\) be a maximal ideal in \(\mathbb {R}[x,y]\) containing \(x^2+y^2+1\). What is the quotient \(\mathbb {R}[x,y]/m\)? Justify your answer.

Suppose \(R\) is a commutative ring with identity. Let \(J\) denote the intersection of all maximal ideals in \(R\). Suppose \(M\) is a finitely-generated \(R\)-module such that \[ JM:=\{jm\mid j\in J\text { and }m\in M\}=\{0\}. \] Prove that \(M=\{0\}\).

Let \(S\) be a multiplicative subset of commutative ring \(A\) not containing 0. Let \(P\) be a maximal element in the set of ideals of \(A\) whose intersection with \(S\) is empty. Show that \(P\) is prime.

Let \((R,m)\) be a local ring (i.e. a commutative ring with a unique maximal ideal \(m\)). Let \(M\) be a finitely generated \(R\)-module with \(mM=M\). Show from first principles that \(M=\{0\}\).

Let \(R\) be the ring of polynomials in one variable over a field \(F\). Prove that \(R\) has infinitely many maximal ideals.

Let \(R\) be a an algebra over a field \(k\) of dimension \(n\). Let \(I\) be the intersection of all maximal left ideals of \(R\). Show that \(I\) is a 2-sided ideal and that \(I^n=0\). Give an example for which \(I^{n-1}\ne 0\).

1. Let \(H\) be a maximal subgroup of the finite group \(G\) and let \(M\) be a minimal normal subgroup of \(G\). Assume that \(M\) is abelian but \(H\) does not contain \(M\). Prove that \(G=HM\), \(H\cap M=\{1\}\) (the identity subgroup), and \(|G:H|\) (the index of \(H\) in \(G\)) is a power of a prime.
2. Use the result in (a) to show that that if \(G\) is a finite solvable group and \(H\) is a maximal subgroup of \(G\), then \(|G:H|\) is a power of a prime.