Cyclic Module Decompositions

Let \(R := \mathbb {Z}\), and let \(M\) be the cokernel of the map \[ R^4 \xrightarrow { \begin {pmatrix} 2 & 7 & 3 & 4\\ 3 & 8 & 7 & 11\\ 4 & 10 & 10 & 16 \end {pmatrix} } R^3. \] Write \(M\) as a direct sum of cyclic \(R\)-modules.

Let \(k=\mathbb {Z}/2\mathbb {Z}\) and \(R=k[x]\). Let \(M\) denote the cokernel of the map \(R^3\to R^3\) given by the matrix: \[ \begin {bmatrix} 1+x&x^2&1+x\\ 1&x&1\\ 1&x^2&1+x^2 \end {bmatrix} \]

Write \(\operatorname {Hom}(M,R/(x))\) as a direct sum of cyclic modules.

Let \(k=\mathbb {Z}/2\mathbb {Z}\) be the field with two elements and let \(R=k[x]\). Let \(M\) be the cokernel of the mapping from \(R^3\) to \(R^3\) given by the matrix \[ \begin {bmatrix} x^2&x^2&0\\ x^2&x^2+x&x+1\\ 0&0&x+1 \end {bmatrix} \]

How many elements are in \(\operatorname {Hom}_R(k[x]/(x),M)\)?

Let \(R:=\mathbb {Q}[x]\), and let \(M\) be the cokernel of the map \[ R^3\xrightarrow {\begin {pmatrix}x-2&4&x\\0&4&x\\0&-x&-1\end {pmatrix}}R^3. \]

Write \(M\) as a direct sum of cyclic \(R\)-modules.

Let \(M\) be the cokernel of the map \[ \mathbb {Z}^3\xrightarrow {\begin {pmatrix}2&2&4\\2&4&6\\2&6&8\end {pmatrix}}\mathbb {Z}^3. \]

Write \(M\) as a direct sum of cyclic \(\mathbb {Z}\)-modules.

Suppose \(R=\mathbb {F}_3[x]\). Let \(M\) be the module generated by three elements \(a,b,c\) subject to the three relations \(-xa+x^2b+(x^2-1)c=0\), \(xb+xc=0\) and \(xb+x^2c=0\). Find a direct sum of cyclic \(R\)-modules isomorphic to \(M\).

Consider \(R:=\mathbb {Q}[x]\) and the matrix: \[ A= \begin {bmatrix} x-1 & 0\\ 1-x & x^2 \end {bmatrix}. \] Let \(M\) be the cokernel of the map \(R^2\longrightarrow R^2\) given by multiplication by \(A\). Compute the rank of the \(\mathbb {Q}\)-vector space \(\operatorname {Hom}_R(M,R/(x^2))\).

Let \(R=\mathbb {F}_3[x]\). Consider the \(3\times 3\) matrix \[ \begin {bmatrix} 0 & x & x\\ 1 & x & 1+x\\ 1+x^2 & x^2 & 1+x^2 \end {bmatrix} \] Let \(\varphi :R^3\longrightarrow R^3\) be the \(R\)-module homomorphism given by this matrix. Write the cokernel of \(\varphi \) as a direct sum of cyclic modules.

Let \(R=\mathbb {Q}[x]\) and consider the submodule \(M\) of \(R^2\) generated by the elements \((x^2-1,x-1)\) and \((x^2+x,x)\). Write \(M\) as a direct sum of cyclic modules.

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.

Let \(R=\mathbb {Q}[x]\), and let \(M\) be the cokernel of the map \[ R^2\longrightarrow R^3 \qquad \text {given by}\qquad \begin {pmatrix} x+1 & 0\\ 2x & x^2\\ 1 & 1 \end {pmatrix}. \] Write \(M\) as a direct sum of cyclic \(R\)-modules.

Let \(R=\mathbb {Q}[x]\), and let \(M\) be the cokernel of the map \(R^2\longrightarrow R^3\) given by the matrix \[ \begin {pmatrix} x & 0\\ x & x^2\\ 1 & 1 \end {pmatrix}. \] Write \(M\) as a direct sum of cyclic \(R\)-modules.

Let \(M\) be the cokernel of the map \(\mathbb {Z}^2\longrightarrow \mathbb {Z}^3\) given by \[ \begin {pmatrix} 3 & 6\\ 4 & 10\\ 10 & 22 \end {pmatrix}. \] Write \(M\) as a direct sum of cyclic groups.

Set \(R=\mathbb {Q}[x]\), and consider the submodule \(M\) of \(R^2\) generated by the elements \((1-2x,-x^2)\) and \((1-x,x-x^2)\). Express \(R^2/M\) as a direct sum of cyclic modules.

Let \(R\) be a ring and let \(M\) be an irreducible \(R\)-module. Prove that one of the following statements is true about \(M\) considered as an abelian group:

1. \(M\) is the direct sum of (perhaps infinitely many) copies of the additive group of all rational numbers.
2. \(M\) is the direct sum of (perhaps infinitely many) copies of a cyclic group of order \(p\) for some prime \(p\).

Let \(R\) be a ring and let \(M\) be an irreducible \(R\)-module. Prove that one of the following statements is true about \(M\) considered as an abelian group:

1. \(M\) is the direct sum of (perhaps infinitely many) copies of the additive group of all rational numbers.
2. \(M\) is the direct sum of (perhaps infinitely many) copies of a cyclic group of order \(p\) for some prime \(p\).