Modules over PIDs

Determine all the \(\mathbb {Z}\)-submodules of the \(\mathbb {Z}\)-module \(\mathbb {Z}/(3)\oplus \mathbb {Z}/(3)\oplus \mathbb {Z}/(9)\) consisting of 3 elements.

Suppose \(R\) is a commutative ring with 1 and \(x, y \in R\) satisfy \(xy = 0\) and \((x, y) = R\) (i.e. the ideal generated by \(x\) and \(y\) is the whole ring). Prove that the \(R\)-module homomorphism

sending \(f\in R\) to its image in \(R/(x)\), and in \(R/(y)\), is an isomorphism.

Let \(R\) be a commutative ring with identity, and let \(0\to L\to M\to N\to 0\) be an exact sequence of \(R\)- modules. Prove or disprove:

1. If \(M\) and \(N\) are finitely generated \(R\)-modules, then \(L\) is finitely generated.
2. If \(M\) is finitely generated, and \(N\) is a free \(R\)-module, then \(L\) is finitely generated.

Prove that the additive group of \(\mathbb {Q}\) is not a projective \(\mathbb {Z}\)-module.

Let \(R=\mathbb {Z}[i]\) be the ring of Gaussian integers. Consider the \(R\)-module \(M\) generated by two elements \(x\) and \(y\), subject to the relations \(ix+2y=0\) and \(2x-iy=0\). How many elements does \(M\) have?

Suppose that \(R=\mathbb {C}[x]\), and that \(M\) is the \(R\)-module generated by three elements \(a,b\), and \(c\), modulo the three relations \(a-xc\), \(xa-xb+xc\), and \(xb-(x^2-1)c\). Write \(M\) as a direct sum of cyclic modules.

Let \(M\) be the \(\mathbb {C}[x]\)-module generated by elements \(a,b,c\) modulo the three relations \(a+b\), \(x^3a+xb+xc\) and \((x^3+1)a+b\). Write \(M\) as a direct sum of cyclic \(\mathbb {C}[x]\)-modules.

Prove that the \(\mathbb {Z}\)-module \(\mathbb {Q}\) is not projective.

Which of the following is a principal ideal domain? (a) \(\mathbb {Z}[i]\), (b) \(\mathbb {Z}[2\sqrt {2}]\), (c) \(\mathbb {Q}[3\sqrt {3}]\).

Prove that the \(\mathbb {Z}\)-module \(\mathbb {Q}\) is not projective.

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.

In the category of \(\mathbb {Z}\)-modules, is the module \(\mathbb {Q}/\mathbb {Z}\) (a) projective? (b) injective? (c) flat?

Justify your answer.

Let \(R\) be a commutative ring with 1. Show that every \(R\)-module is free if and only if \(R\) is a field.

Describe all finitely generated \(\mathbb {Z}\)-submodules of \(\mathbb {Q}\).

Let \(M\) be a \(\mathbb {Z}\)-module consisting of \(3\times 1\)-matrics with integer entries. Let \(N\subseteq M\) be a \(\mathbb {Z}\)-submodule generated by the columns of the matrix \[ \begin {pmatrix} 2 & 7 & 3 & 4\\ 3 & 8 & 7 & 11\\ 4 & 10 & 10 & 16 \end {pmatrix}. \] Find a minimal presentation for the quotient \(M/N\).

Let \(R\) be a commutative integral domain with identity. Recall that an \(R\)-module is torsion free if for any \(r\in R\) and \(m\in M\), \(rm=0\) implies that either \(r=0\) or \(m=0\).

1. Suppose \(R\) is a principal ideal domain and \(M\) and \(N\) are two \(R\) modules. Prove that if \(M\) and \(N\) are torsion free, so is \(M\otimes _R N\).
2. Give an example of an integral domain \(R\) and two torsion free \(R\)-modules \(M\) and \(N\) such that \(M\otimes _R N\) is not torsion free.

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 \((R,m)\) 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 \(M=\{0\}\).

Prove that a principal ideal domain is a unique factorization domain.

Show that the endomorphism ring of a simple module is a division ring.

Prove Wedderburn’s theorem: if the ring \(R\) has a faithful simple module that is finite-dimensional over its commutant ring, then \(R\cong M_n(D)\) for some \(n\ge 1\), where \(D\) is a division ring.

Show that if a module \(M\) is a sum of simple modules, then it is a direct sum of simple modules.

1. Define the elementary divisors of a square matrix \(M\) over the principal ideal domain \(R\).
2. Calculate the elementary divisors of the matrix of integers \[ \begin {pmatrix} 4 & 6\\ 6 & 12 \end {pmatrix}. \]

1. Let \(R\) be a ring and let \(M\) be an \(R\)-module. Prove that if \(M\) is generated by simple submodules, then \(M\) is the direct sum of simple submodules.
2. Show that every vector space over a field has a basis.

If \(R\) is a ring and \(M\) is a simple \(R\)-module, then \(\operatorname {End}_R(M)\) is a division ring.