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Exercise 2.1.6 (i) of Berrick and Keating's book An Introduction to Rings and Modules reads as follows:Let $$M = M_1 \oplus M_2$$, an internal sum of right $$R$$-modules, and let $$\{ \sigma_1 , \sigma_2, \pi_1 , \pi_2 \}$$ be the corresponding set of inclusions and projections.
Given an endomorphism $$\mu$$ of $$M$$, define $$\mu_{ij} = \pi_i \mu \sigma_j$$, an $$R$$-homomorphism from $$M_j$$ to $$M_i, i,j = 1,2$$.Show that for $$m = m_1 + m_2$$, with $$m_1 \in M_1, m_2 \in M_2$$ we have:
$$\mu(m) = ( \mu_{11}m_1 + \mu_{12}m_2) + ( \mu_{21}m_1 + \mu_{22}m_2)
$$
where $$\mu_{11}m_1 + \mu_{12}m_2$$ is in $$M_1$$
and
$$\mu_{21}m_1 + \mu_{22}m_2$$ is in $$M_2$$
Viewing $$M$$ as a 'column space' $$\begin{bmatrix} M_1 \\ M_2 \end{bmatrix}$$, show that $$\mu$$ can be represented as a matrix $$\begin{bmatrix}\mu_{11} & \mu_{12} \\ \mu_{21} & \mu_{22} \end{bmatrix}.$$
Deduce that the ring of endomorphisms End($$M$$) of $$M$$ can be written as a ring of $$2 \times 2$$ matrices:
$$End(M) = \begin{bmatrix} End(M_1) & Hom(M_2, M_1) \\ Hom(M_1, M_2) & End(M_2) \end{bmatrix}$$
where $$Hom(M_1, M_2)$$ is the set of all $$R$$-Module maps from $$M_1$$ to $$M_2$$.
Can someone please help me to get started on this exercise.
Help will be appreciated!
Peter
Given an endomorphism $$\mu$$ of $$M$$, define $$\mu_{ij} = \pi_i \mu \sigma_j$$, an $$R$$-homomorphism from $$M_j$$ to $$M_i, i,j = 1,2$$.Show that for $$m = m_1 + m_2$$, with $$m_1 \in M_1, m_2 \in M_2$$ we have:
$$\mu(m) = ( \mu_{11}m_1 + \mu_{12}m_2) + ( \mu_{21}m_1 + \mu_{22}m_2)
$$
where $$\mu_{11}m_1 + \mu_{12}m_2$$ is in $$M_1$$
and
$$\mu_{21}m_1 + \mu_{22}m_2$$ is in $$M_2$$
Viewing $$M$$ as a 'column space' $$\begin{bmatrix} M_1 \\ M_2 \end{bmatrix}$$, show that $$\mu$$ can be represented as a matrix $$\begin{bmatrix}\mu_{11} & \mu_{12} \\ \mu_{21} & \mu_{22} \end{bmatrix}.$$
Deduce that the ring of endomorphisms End($$M$$) of $$M$$ can be written as a ring of $$2 \times 2$$ matrices:
$$End(M) = \begin{bmatrix} End(M_1) & Hom(M_2, M_1) \\ Hom(M_1, M_2) & End(M_2) \end{bmatrix}$$
where $$Hom(M_1, M_2)$$ is the set of all $$R$$-Module maps from $$M_1$$ to $$M_2$$.
Can someone please help me to get started on this exercise.
Help will be appreciated!
Peter
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