MHB Endomorphisms of Direct Sums - Berrick and Keating - Exercise 2.1.6 (i)

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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
 
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Peter said:
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

Suppose $\mu(m) = \mu(m_1 + m_2) = n_1 + n_2$ with $n_1 \in M_1,n_2 \in M_2$.

Now $\mu_{11}(m_1) = \pi_1\mu\sigma_1(m_1) = \pi_1\mu(m_1 + 0)$

$\mu_{12}(m_2) = \pi_1\mu\sigma_2(m_2) = \pi_1\mu(0 + m_2)$

$\mu_{21}(m_1) = \pi_2\mu\sigma_1(m_1) = \pi_2\mu(m_1 + 0)$

$\mu_{22}(m_2) = \pi_2\mu\sigma_2(m_2) = \pi_2\mu(0+m_2)$.

Thus:

$(\mu_{11}(m_1) + \mu_{12}(m_2)) + (\mu_{21}(m_1) + \mu_{22}(m_2))$

$= (\pi_1\mu(m_1 + 0) + \pi_1\mu(0 + m_2)) + (\pi_2\mu(m_1 + 0) + \pi_2\mu(0+m_2))$

$= (\pi_1\mu((m_1 + 0) + (0 + m_2)) + (\pi_2\mu((m_1 + 0) + (0 + m_2))$

$= (\pi_1\mu(m_1 + m_2)) + (\pi_2\mu(m_1 + m_2))$

$= (\pi_1\mu(m)) + (\pi_2\mu(m)) = \pi_1(n_1+n_2) + \pi_2(n_1 + n_2)$

$ = n_1 + n_2 = \mu(m)$.

Can you take it from here?
 
Deveno said:
Suppose $\mu(m) = \mu(m_1 + m_2) = n_1 + n_2$ with $n_1 \in M_1,n_2 \in M_2$.

Now $\mu_{11}(m_1) = \pi_1\mu\sigma_1(m_1) = \pi_1\mu(m_1 + 0)$

$\mu_{12}(m_2) = \pi_1\mu\sigma_2(m_2) = \pi_1\mu(0 + m_2)$

$\mu_{21}(m_1) = \pi_2\mu\sigma_1(m_1) = \pi_2\mu(m_1 + 0)$

$\mu_{22}(m_2) = \pi_2\mu\sigma_2(m_2) = \pi_2\mu(0+m_2)$.

Thus:

$(\mu_{11}(m_1) + \mu_{12}(m_2)) + (\mu_{21}(m_1) + \mu_{22}(m_2))$

$= (\pi_1\mu(m_1 + 0) + \pi_1\mu(0 + m_2)) + (\pi_2\mu(m_1 + 0) + \pi_2\mu(0+m_2))$

$= (\pi_1\mu((m_1 + 0) + (0 + m_2)) + (\pi_2\mu((m_1 + 0) + (0 + m_2))$

$= (\pi_1\mu(m_1 + m_2)) + (\pi_2\mu(m_1 + m_2))$

$= (\pi_1\mu(m)) + (\pi_2\mu(m)) = \pi_1(n_1+n_2) + \pi_2(n_1 + n_2)$

$ = n_1 + n_2 = \mu(m)$.

Can you take it from here?
Thanks for the start Deveno ... gives me confidence to continue ...

Will try to progress the solution a bit ...

Peter
 
Deveno said:
Suppose $\mu(m) = \mu(m_1 + m_2) = n_1 + n_2$ with $n_1 \in M_1,n_2 \in M_2$.

Now $\mu_{11}(m_1) = \pi_1\mu\sigma_1(m_1) = \pi_1\mu(m_1 + 0)$

$\mu_{12}(m_2) = \pi_1\mu\sigma_2(m_2) = \pi_1\mu(0 + m_2)$

$\mu_{21}(m_1) = \pi_2\mu\sigma_1(m_1) = \pi_2\mu(m_1 + 0)$

$\mu_{22}(m_2) = \pi_2\mu\sigma_2(m_2) = \pi_2\mu(0+m_2)$.

Thus:

$(\mu_{11}(m_1) + \mu_{12}(m_2)) + (\mu_{21}(m_1) + \mu_{22}(m_2))$

$= (\pi_1\mu(m_1 + 0) + \pi_1\mu(0 + m_2)) + (\pi_2\mu(m_1 + 0) + \pi_2\mu(0+m_2))$

$= (\pi_1\mu((m_1 + 0) + (0 + m_2)) + (\pi_2\mu((m_1 + 0) + (0 + m_2))$

$= (\pi_1\mu(m_1 + m_2)) + (\pi_2\mu(m_1 + m_2))$

$= (\pi_1\mu(m)) + (\pi_2\mu(m)) = \pi_1(n_1+n_2) + \pi_2(n_1 + n_2)$

$ = n_1 + n_2 = \mu(m)$.

Can you take it from here?
Hi Deveno ... just a minor clarification:

You write:

" ... ... Suppose $\mu(m) = \mu(m_1 + m_2) = n_1 + n_2$ with $n_1 \in M_1,n_2 \in M_2$. ... ... "

How can we be sure that $$m$$ mapped under $$\mu$$ gives us (neatly) one element from $$M_1$$ and one element from $$M_2$$?

Is it because $$\mu (m) = n$$ for some $$n \in M$$ and given the nature of the direct sum we have $$n = n_1 + n_2$$ where $$n_1 \in M_1$$ and $$n_2 \in M_2$$? Can you confirm that this is the reason?

Second clarification:

If we replaced $$(m_1 + m_2)$$ by $$(m_1, m_2)$$ in the proof/solution , would everything still hold? [I am thinking that given the nature of direct sums this would be OK.]

Peter
 
Deveno said:
Suppose $\mu(m) = \mu(m_1 + m_2) = n_1 + n_2$ with $n_1 \in M_1,n_2 \in M_2$.

Now $\mu_{11}(m_1) = \pi_1\mu\sigma_1(m_1) = \pi_1\mu(m_1 + 0)$

$\mu_{12}(m_2) = \pi_1\mu\sigma_2(m_2) = \pi_1\mu(0 + m_2)$

$\mu_{21}(m_1) = \pi_2\mu\sigma_1(m_1) = \pi_2\mu(m_1 + 0)$

$\mu_{22}(m_2) = \pi_2\mu\sigma_2(m_2) = \pi_2\mu(0+m_2)$.

Thus:

$(\mu_{11}(m_1) + \mu_{12}(m_2)) + (\mu_{21}(m_1) + \mu_{22}(m_2))$

$= (\pi_1\mu(m_1 + 0) + \pi_1\mu(0 + m_2)) + (\pi_2\mu(m_1 + 0) + \pi_2\mu(0+m_2))$

$= (\pi_1\mu((m_1 + 0) + (0 + m_2)) + (\pi_2\mu((m_1 + 0) + (0 + m_2))$

$= (\pi_1\mu(m_1 + m_2)) + (\pi_2\mu(m_1 + m_2))$

$= (\pi_1\mu(m)) + (\pi_2\mu(m)) = \pi_1(n_1+n_2) + \pi_2(n_1 + n_2)$

$ = n_1 + n_2 = \mu(m)$.

Can you take it from here?
Just another minor clarification, Deveno ... ...

In your working above, you put

$$\pi_1\mu(m_1 + 0) + \pi_1\mu(0 + m_2) $$

equal to

$$ \pi_1\mu((m_1 + 0) + (0 + m_2)) $$

Is the justification for this that $$\pi_1$$ is a module homomorphism, $$\mu$$ is also and so their composition is a module homomorphism?

Peter
Peter
 
Peter said:
Hi Deveno ... just a minor clarification:

You write:

" ... ... Suppose $\mu(m) = \mu(m_1 + m_2) = n_1 + n_2$ with $n_1 \in M_1,n_2 \in M_2$. ... ... "

How can we be sure that $$m$$ mapped under $$\mu$$ gives us (neatly) one element from $$M_1$$ and one element from $$M_2$$?

Is it because $$\mu (m) = n$$ for some $$n \in M$$ and given the nature of the direct sum we have $$n = n_1 + n_2$$ where $$n_1 \in M_1$$ and $$n_2 \in M_2$$? Can you confirm that this is the reason?

Second clarification:

If we replaced $$(m_1 + m_2)$$ by $$(m_1, m_2)$$ in the proof/solution , would everything still hold? [I am thinking that given the nature of direct sums this would be OK.]

Peter

We have an INTERNAL direct sum so any element $m \in M$ has a UNIQUE expression as $m_1 + m_2$. This also holds for the element $\mu(m) \in M$.

So, yes, you are correct.

The "side by side" writing of $m_1 + m_2$ as $(m_1,m_2)$ is for an EXTERNAL direct product. We don't need to do that, here.
Peter said:
Just another minor clarification, Deveno ... ...

In your working above, you put

$$\pi_1\mu(m_1 + 0) + \pi_1\mu(0 + m_2) $$

equal to

$$ \pi_1\mu((m_1 + 0) + (0 + m_2)) $$

Is the justification for this that $$\pi_1$$ is a module homomorphism, $$\mu$$ is also and so their composition is a module homomorphism?

Peter
Peter

Yep.
 
Deveno said:
We have an INTERNAL direct sum so any element $m \in M$ has a UNIQUE expression as $m_1 + m_2$. This also holds for the element $\mu(m) \in M$.

So, yes, you are correct.

The "side by side" writing of $m_1 + m_2$ as $(m_1,m_2)$ is for an EXTERNAL direct product. We don't need to do that, here.

Yep.

Thanks Deveno ... hmmm ... missed the 'unique' bit ... thanks for the reminder ...

Peter
 
Deveno said:
Suppose $\mu(m) = \mu(m_1 + m_2) = n_1 + n_2$ with $n_1 \in M_1,n_2 \in M_2$.

Now $\mu_{11}(m_1) = \pi_1\mu\sigma_1(m_1) = \pi_1\mu(m_1 + 0)$

$\mu_{12}(m_2) = \pi_1\mu\sigma_2(m_2) = \pi_1\mu(0 + m_2)$

$\mu_{21}(m_1) = \pi_2\mu\sigma_1(m_1) = \pi_2\mu(m_1 + 0)$

$\mu_{22}(m_2) = \pi_2\mu\sigma_2(m_2) = \pi_2\mu(0+m_2)$.

Thus:

$(\mu_{11}(m_1) + \mu_{12}(m_2)) + (\mu_{21}(m_1) + \mu_{22}(m_2))$

$= (\pi_1\mu(m_1 + 0) + \pi_1\mu(0 + m_2)) + (\pi_2\mu(m_1 + 0) + \pi_2\mu(0+m_2))$

$= (\pi_1\mu((m_1 + 0) + (0 + m_2)) + (\pi_2\mu((m_1 + 0) + (0 + m_2))$

$= (\pi_1\mu(m_1 + m_2)) + (\pi_2\mu(m_1 + m_2))$

$= (\pi_1\mu(m)) + (\pi_2\mu(m)) = \pi_1(n_1+n_2) + \pi_2(n_1 + n_2)$

$ = n_1 + n_2 = \mu(m)$.

Can you take it from here?

You write:

" ... ... Can you take it from here? ... ..."

Well, the second part of the exercise looks pretty straightforward ... unless I am misinterpreting it ...

The second part of the exercise states:

"Viewing $$M$$ as a 'column space' $$\begin{bmatrix} M_1 \\ M_2 \end{bmatrix}$$, show that $$\mu$$ can be represented as a matrix $$\begin{bmatrix}\mu_{11} & \mu_{12} \\ \mu_{21} & \mu_{22} \end{bmatrix}.$$"

If we take $$m = \begin{bmatrix} m_1 \\ m_2 \end{bmatrix} $$

where $$m_1 \in M_1$$ and $$m_2 \in M_2$$

then we can represent $$\mu(m_1)$$ as follows:

$$\mu (m_1) = \begin{bmatrix}\mu_{11} & \mu_{12} \\ \mu_{21} & \mu_{22} \end{bmatrix} \begin{bmatrix} m_1 \\ m_2 \end{bmatrix} $$

$$= \begin{bmatrix} \mu_{11}m_1 + \mu_{12}m_2 \\ \mu_{21}m_1 + \mu_{22}m_2 \end{bmatrix}$$

where $$\mu_{11}m_1 + \mu_{12}m_2 \in M_1$$ and $$\mu_{21}m_1 + \mu_{22}m_2 \in M_2$$

Can you confirm that this is a satisfactory answer to this part of the exercise.

Peter
 
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