MHB Infinite Direct Sums and Indexed Sets

[Math Amateur]

I am reading Chapter 2: Vector Spaces over $$\mathbb{Q}, \mathbb{R} \text{ and } \mathbb{C}$$ of Anthony W. Knapp's book, Basic Algebra.

I need some help with some issues regarding infinite direct sums and products and indexed families of sets ... ...

On page 62, Knapp introduces direct sums and products of infinitely many vector spaces as follows:View attachment 2935So ... , in the above text Knapp introduces the direct sum of infinitely many vector spaces as follows:Let A be some index set.

Then the direct sum, $$ \bigoplus_{\alpha \in A} V_\alpha $$ is the set of all tuples $$\{ v_\alpha \}$$ in the Cartesian product $$X_{\alpha \in A} $$ with all but finitely many $$v_\alpha$$ equal to zero and with addition and multiplication defined co-ordinate by co-ordinate ... ... "
My questions and my puzzlement in this matter relates to the nature of the tuples $$v_\alpha$$ ... ... what is their nature and properties ...?
I have a series of questions as follows:Firstly, ... just clarifying the notation, and indeed the nature of the tuple ... ... is the tuple $$\{ v_\alpha \}$$ ... ... or is the tuple actually$$ v_\alpha$$ ... ... whereas $$\{ v_\alpha \}$$ is the set of all tuples ...Secondly can we somehow write the following:

$$ (v_\alpha) = ( v_{\alpha_1} , v_{\alpha_2}, v_{\alpha_3}, ... \ ... , v_{\alpha_n} , ... \ ... ) $$

where $$\alpha_i \in A$$Thirdly, does the index set have to have some order or ordering relation?Fourthly, does the tuple $$v_\alpha$$ (or should I write $$(v_\alpha) )$$ have some kind of order?Fifth and last question ... ... Can the index set be an uncountably infinite set? If so what is the nature of the tuple $$v_\alpha $$and can/does its coordinates have some kind of order ...Hoping someone can answer these questions ...

Further, can someone give some simple examples, including one involving an uncountably infinite index set? That would really help ... ...

Hoping that someone can help ... ...

Peter

 

[Euge]

Firstly, ... just clarifying the notation, and indeed the nature of the tuple ... ... is the tuple $$\{ v_\alpha \}$$ ... ... or is the tuple actually$$ v_\alpha$$ ... ... whereas $$\{ v_\alpha \}$$ is the set of all tuples ...

The $(v_\alpha)$ (which are the elements of a direct sum or product) are tuples, and the $v_\alpha$ in $(v_\alpha)$ are coordinates. Recall from calculus (or analysis) that elements of $\Bbb R^n$ are $n$-tuples and in an $n$-tuple $(a_1,\ldots, a_n)$, $a_i$ is the i-th coordinate. The ideas introduced by Knapp are things you've probably seen before.

Secondly can we somehow write the following:

$$ (v_\alpha) = ( v_{\alpha_1} , v_{\alpha_2}, v_{\alpha_3}, ... \ ... , v_{\alpha_n} , ... \ ... ) $$

where $$\alpha_i \in A$$

That would be acceptable if the index set is countable. However, if the index set is uncountable, you cannot write $(v_\alpha)$ as the sequence you have.

Thirdly, does the index set have to have some order or ordering relation?

That is not required in the definition. But what is important here is what equality means for tuples; $(u_\alpha) = (v_\alpha)$ if and only if $u_\alpha = v_\alpha$ for all $\alpha$. In other words, if $u = (u_\alpha)$ and $v = (v_\alpha)$, then $u = v$ if and only if $\pi_\alpha(u) = \pi_\alpha(v)$ for all $\alpha$ (the $\pi_\alpha$ are the natural projection maps onto $V_\alpha$). This tells you implicitly that order is important in some sense. For example, in $\Bbb R^2$, the pair $(1,2)$ is not equal to $(2,1)$ (Do you remember the term "ordered pair"?)

In case you were thinking of nets in topology, where the index set needs to satisfy a certain partial ordering, here's the thing. Although nets are tuples, the coordinates need not come from a vector space. We're dealing with vector spaces here.

Fourthly, does the tuple $$v_\alpha$$ (or should I write $$(v_\alpha) )$$ have some kind of order?

What do you mean by that?

Fifth and last question ... ... Can the index set be an uncountably infinite set? If so what is the nature of the tuple $$v_\alpha $$and can/does its coordinates have some kind of order ...

Yes, as I have mentioned before, the index set can be uncountable. I'm not sure what you mean by "the nature of the tuples," but any properties the tuples have must come from the respective vector spaces that their coordinates lie in. This is certainly the case no matter the size of the index set. For example, in both $\Bbb F_2 \times \Bbb F_2$ and $\prod_{\alpha\in [0,1]} \Bbb F_2$, $2x = 0$ for all $x$.

Further, can someone give some simple examples, including one involving an uncountably infinite index set? That would really help ... ...

Sure. Since you're just getting used to the material, I'll avoid using sheaves and categorical constructions.

Example 1. Let $\Bbb R^\infty$ be the set of sequences $(a_n)$ of real numbers. Then $\Bbb R^\infty = \prod_{n = 1}^\infty \Bbb R$. Note that $\Bbb R^\infty$ is not $\oplus_{n = 1}^\infty \Bbb R$ since the constant sequence $(1,1, 1,\ldots)$ is not the direct sum.

Example 2. Let $\{V_ \alpha\}_{\alpha\in A}$ be a collection of vector spaces over a field $\Bbb k$ ($A$ can be uncountable). Let $V$ be any vector space over $k$. There are group isomorphisms

$\displaystyle \text{Hom}_{\Bbb k}\left(V, \prod_{\alpha\in A} V_\alpha\right) \cong \prod_{\alpha \in A} \text{Hom}(V, V_\alpha) $

and

$\displaystyle \text{Hom}_{\Bbb k}\left(\oplus_{\alpha\in A} V_\alpha, V\right) \cong \prod_{\alpha\in A} \text{Hom}_k(V_\alpha, V)$.

Notice where the direct sum and product are placed inside $\text{Hom}_{\Bbb k}$. The first isomorphism is obtained by the map

$\displaystyle R : \text{Hom}_{\Bbb k}(V, \prod_{\alpha\in A} V_\alpha) \to \prod_{\alpha\in A} \text{Hom}_{\Bbb k}(V, V_\alpha)$

which sends $f$ to the tuple $(f_\alpha)$, where $f_\alpha(v) = (\pi_{\alpha}(f(v)))$. Its inverse is the map $L : (f_\alpha) \to \prod_{\alpha} f_{\alpha}$, where $\prod_{\alpha} f_\alpha(v) = (f_\alpha)(v)$.

The second isomorphism is obtained via the map

$S : \text{Hom}_{\Bbb k}\left(\oplus_{\alpha\in A} V_{\alpha}, V\right) \to \prod_{\alpha\in A} \text{Hom}_{\alpha}(V_\alpha, V)$

which sends $f$ to $\prod_{\alpha\in A} f\circ \iota_{\alpha}$ (the $\iota_\alpha$ are inclusion maps from $ V_\alpha$ to $\oplus_{\alpha\in A} V_{\alpha}$). Its inverse is the map $T : (f_\alpha) \mapsto \sum_{\alpha\in A} f_\alpha$.

Example 3. Let $M$ be a (real) smooth manifold of dimension $n$. Given $p\in M$, let $T_p(M)$ be the vector space of tangent vectors to $p$ in $M$, and let $T_p^*(M)$ denote the cotangent space to $p$ in $M$ (i.e. the space of linear functionals on $T_p(M)$). A differential $k$-form on $M$ assigns to every $p\in M$, an alternating linear map $\nu(p) : (T_p^*(M))^k \to \Bbb R$. In $\Bbb R^3$ for example, the map $dx \wedge dy: (v,w) \mapsto v_1 w_2 - v_2 w_1$ is a differential 2-form.

Loosely speaking, direct sums are made from inclusion maps and direct products are made from natural projections. These arrows (inclusions and projections) are in reverse directions. These concepts are made more precise with the universal properties of direct sums and products, which you'll see later.

 

[Deveno]

The tuple is $\{v_{\alpha}\}$. $v_{\alpha}$ is the $\alpha$-th "coordinate".

With a FINITE index set, which may as well be $\{1,2,\dots,n\}$, we can just LIST these coordinates:

$(v_1,v_2,\dots,v_n)$.

With an infinite index set (which may be uncountable), it's not always possible to do this. So we need another way to describe an "infinite-tuple".

Formally, a tuple is a mapping:

$x:A \to X$, where $x(\alpha)$ is typically denoted $x_{\alpha}$.

To give an example of what an uncountably indexed family might look like, consider the open interval:

$(0,x) \subseteq \Bbb R$, for $x \in \Bbb R^+$. It is clear we get an open interval like this for any positive real number, and the collection of all these intervals can be regarded as an indexed set:

$\{I_x\}_{x \in \Bbb R}$, where $I_x = (0,x)$.

To answer your second question-no, we generally CANNOT write:

$(v_{\alpha}) = (v_{\alpha_1},v_{\alpha_2},v_{\alpha_3},\dots,v_{\alpha_n},\dots)$

Because there is typically no clear way to list:

$\{\alpha_1,\alpha_2,\dots,\alpha_n,\dots\}$.

While it IS a theorem of ZFC that any set can be well-ordered, an explicit description of said well-ordering is often impossible to do. And even if it COULD be described, it could not be written down in a list fashion, if the set is uncountable (for writing such a list that eventually listed "every element" would establish a bijection with the natural numbers, and thus means the set WAS countable).

Your questions actually touch on some deep areas of set theory...and our normal intuitions are not much help, there. Sets can be "very big" and so index sets can likewise be "very big".

 

[Math Amateur]

The $(v_\alpha)$ (which are the elements of a direct sum or product) are tuples, and the $v_\alpha$ in $(v_\alpha)$ are coordinates. Recall from calculus (or analysis) that elements of $\Bbb R^n$ are $n$-tuples and in an $n$-tuple $(a_1,\ldots, a_n)$, $a_i$ is the i-th coordinate. The ideas introduced by Knapp are things you've probably seen before.
That would be acceptable if the index set is countable. However, if the index set is uncountable, you cannot write $(v_\alpha)$ as the sequence you have.
That is not required in the definition. But what is important here is what equality means for tuples; $(u_\alpha) = (v_\alpha)$ if and only if $u_\alpha = v_\alpha$ for all $\alpha$. In other words, if $u = (u_\alpha)$ and $v = (v_\alpha)$, then $u = v$ if and only if $\pi_\alpha(u) = \pi_\alpha(v)$ for all $\alpha$ (the $\pi_\alpha$ are the natural projection maps onto $V_\alpha$). This tells you implicitly that order is important in some sense. For example, in $\Bbb R^2$, the pair $(1,2)$ is not equal to $(2,1)$ (Do you remember the term "ordered pair"?)

In case you were thinking of nets in topology, where the index set needs to satisfy a certain partial ordering, here's the thing. Although nets are tuples, the coordinates need not come from a vector space. We're dealing with vector spaces here.

What do you mean by that?
Yes, as I have mentioned before, the index set can be uncountable. I'm not sure what you mean by "the nature of the tuples," but any properties the tuples have must come from the respective vector spaces that their coordinates lie in. This is certainly the case no matter the size of the index set. For example, in both $\Bbb F_2 \times \Bbb F_2$ and $\prod_{\alpha\in [0,1]} \Bbb F_2$, $2x = 0$ for all $x$.
Sure. Since you're just getting used to the material, I'll avoid using sheaves and categorical constructions.

Example 1. Let $\Bbb R^\infty$ be the set of sequences $(a_n)$ of real numbers. Then $\Bbb R^\infty = \prod_{n = 1}^\infty \Bbb R$. Note that $\Bbb R^\infty$ is not $\oplus_{n = 1}^\infty \Bbb R$ since the constant sequence $(1,1, 1,\ldots)$ is not the direct sum.

Example 2. Let $\{V_ \alpha\}_{\alpha\in A}$ be a collection of vector spaces over a field $\Bbb k$ ($A$ can be uncountable). Let $V$ be any vector space over $k$. There are group isomorphisms

$\displaystyle \text{Hom}_{\Bbb k}\left(V, \prod_{\alpha\in A} V_\alpha\right) \cong \prod_{\alpha \in A} \text{Hom}(V, V_\alpha) $

and

$\displaystyle \text{Hom}_{\Bbb k}\left(\oplus_{\alpha\in A} V_\alpha, V\right) \cong \prod_{\alpha\in A} \text{Hom}_k(V_\alpha, V)$.

Notice where the direct sum and product are placed inside $\text{Hom}_{\Bbb k}$. The first isomorphism is obtained by the map

$\displaystyle R : \text{Hom}_{\Bbb k}(V, \prod_{\alpha\in A} V_\alpha) \to \prod_{\alpha\in A} \text{Hom}_{\Bbb k}(V, V_\alpha)$

which sends $f$ to the tuple $(f_\alpha)$, where $f_\alpha(v) = (\pi_{\alpha}(f(v)))$. Its inverse is the map $L : (f_\alpha) \to \prod_{\alpha} f_{\alpha}$, where $\prod_{\alpha} f_\alpha(v) = (f_\alpha)(v)$.

The second isomorphism is obtained via the map

$S : \text{Hom}_{\Bbb k}\left(\oplus_{\alpha\in A} V_{\alpha}, V\right) \to \prod_{\alpha\in A} \text{Hom}_{\alpha}(V_\alpha, V)$

which sends $f$ to $\prod_{\alpha\in A} f\circ \iota_{\alpha}$ (the $\iota_\alpha$ are inclusion maps from $ V_\alpha$ to $\oplus_{\alpha\in A} V_{\alpha}$). Its inverse is the map $T : (f_\alpha) \mapsto \sum_{\alpha\in A} f_\alpha$.

Example 3. Let $M$ be a (real) smooth manifold of dimension $n$. Given $p\in M$, let $T_p(M)$ be the vector space of tangent vectors to $p$ in $M$, and let $T_p^*(M)$ denote the cotangent space to $p$ in $M$ (i.e. the space of linear functionals on $T_p(M)$). A differential $k$-form on $M$ assigns to every $p\in M$, an alternating linear map $\nu(p) : (T_p^*(M))^k \to \Bbb R$. In $\Bbb R^3$ for example, the map $dx \wedge dy: (v,w) \mapsto v_1 w_2 - v_2 w_1$ is a differential 2-form.

Loosely speaking, direct sums are made from inclusion maps and direct products are made from natural projections. These arrows (inclusions and projections) are in reverse directions. These concepts are made more precise with the universal properties of direct sums and products, which you'll see later.

Thanks Euge ... very much appreciate your considerable help ... still reflecting on your Examples 2 and 3 ...

Peter

 

[Math Amateur]

The tuple is $\{v_{\alpha}\}$. $v_{\alpha}$ is the $\alpha$-th "coordinate".

With a FINITE index set, which may as well be $\{1,2,\dots,n\}$, we can just LIST these coordinates:

$(v_1,v_2,\dots,v_n)$.

With an infinite index set (which may be uncountable), it's not always possible to do this. So we need another way to describe an "infinite-tuple".

Formally, a tuple is a mapping:

$x:A \to X$, where $x(\alpha)$ is typically denoted $x_{\alpha}$.

To give an example of what an uncountably indexed family might look like, consider the open interval:

$(0,x) \subseteq \Bbb R$, for $x \in \Bbb R^+$. It is clear we get an open interval like this for any positive real number, and the collection of all these intervals can be regarded as an indexed set:

$\{I_x\}_{x \in \Bbb R}$, where $I_x = (0,x)$.

To answer your second question-no, we generally CANNOT write:

$(v_{\alpha}) = (v_{\alpha_1},v_{\alpha_2},v_{\alpha_3},\dots,v_{\alpha_n},\dots)$

Because there is typically no clear way to list:

$\{\alpha_1,\alpha_2,\dots,\alpha_n,\dots\}$.

While it IS a theorem of ZFC that any set can be well-ordered, an explicit description of said well-ordering is often impossible to do. And even if it COULD be described, it could not be written down in a list fashion, if the set is uncountable (for writing such a list that eventually listed "every element" would establish a bijection with the natural numbers, and thus means the set WAS countable).

Your questions actually touch on some deep areas of set theory...and our normal intuitions are not much help, there. Sets can be "very big" and so index sets can likewise be "very big".

Hi Deveno & Euge,

Thanks to you both for considerable help on this topic ...

I think I am getting the notation, terminology and ideas straight on this ... BUT ... to check I will just set out some terminology and notation and cast Deveno's example into the framework ... could you just confirm that my notation etc is OK and my interpretation of the example is OK ...

------------------------------------------------

Let $$ \mathcal{X} = \{ X_\alpha \}_{\alpha \in \Delta} = \{ X_\alpha | \alpha \in \Delta \} $$ be an indexed family of sets ... ...

In the above construction, as I understand the terminology, the index set $$\Delta$$ can be any (completely arbitrary) set.

$$\Delta$$ is called the indexing set for $$\mathcal{X}$$

The indexing function $$f \ : \ \Delta \to \mathcal{X}$$ is a mapping from $$\Delta$$ to $$\mathcal{X}$$ such that $$f( \alpha ) = X_\alpha$$

Let $$\prod_\Delta X_\alpha$$ denote the Cartesian Product of a family of sets $$\{ X_\alpha \}_\Delta $$

An element$$ (x_\alpha)$$ of $$\prod_\Delta X_\alpha$$ is referred to as a $$\Delta$$-tuple.

Now using the above notation and turning to Deveno's example:

$$X_\alpha = (0, \alpha) \subseteq \mathbb{R}$$ for $$\alpha \in \mathbb{R}^+$$

So, $$\{ X_\alpha \}_{\alpha \in \Delta}$$ is ,in this case, an indexed family of sets - indeed an uncountable but indexed family of sets ...

and

$$\prod_\Delta X_\alpha$$ is a Cartesian Product of an uncountable family of sets.

An element$$ (x_\alpha)$$ of $$\prod_\Delta X_\alpha$$ is a $$\Delta$$-tuple, but obviously it has an uncountable number of elements and we, again obviously, cannot write it out as a set of coordinates.Can you confirm that the above notation, terminology and interpretation makes sense ...

Peter

 

[Euge]

Hi Deveno & Euge,

Thanks to you both for considerable help on this topic ...

I think I am getting the notation, terminology and ideas straight on this ... BUT ... to check I will just set out some terminology and notation and cast Deveno's example into the framework ... could you just confirm that my notation etc is OK and my interpretation of the example is OK ...

------------------------------------------------

Let $$ \mathcal{X} = \{ X_\alpha \}_{\alpha \in \Delta} = \{ X_\alpha | \alpha \in \Delta \} $$ be an indexed family of sets ... ...

In the above construction, as I understand the terminology, the index set $$\Delta$$ can be any (completely arbitrary) set.

$$\Delta$$ is called the indexing set for $$\mathcal{X}$$

The indexing function $$f \ : \ \Delta \to \mathcal{X}$$ is a mapping from $$\Delta$$ to $$\mathcal{X}$$ such that $$f( \alpha ) = X_\alpha$$

Let $$\prod_\Delta X_\alpha$$ denote the Cartesian Product of a family of sets $$\{ X_\alpha \}_\Delta $$

An element$$ (x_\alpha)$$ of $$\prod_\Delta X_\alpha$$ is referred to as a $$\Delta$$-tuple.

Now using the above notation and turning to Deveno's example:

$$X_\alpha = (0, \alpha) \subseteq \mathbb{R}$$ for $$\alpha \in \mathbb{R}^+$$

So, $$\{ X_\alpha \}_{\alpha \in \Delta}$$ is ,in this case, an indexed family of sets - indeed an uncountable but indexed family of sets ...

and

$$\prod_\Delta X_\alpha$$ is a Cartesian Product of an uncountable family of sets.

An element$$ (x_\alpha)$$ of $$\prod_\Delta X_\alpha$$ is a $$\Delta$$-tuple, but obviously it has an uncountable number of elements and we, again obviously, cannot write it out as a set of coordinates.Can you confirm that the above notation, terminology and interpretation makes sense ...

Peter

Hi Peter,

The product $\prod_{\alpha\in \Delta} X_{\alpha}$ is the set of functions $x : \Delta \to \cup_{\alpha\in \Delta} X_\alpha$ such that for each $\alpha$, $x(\alpha) \in X_{\alpha}$. The elements of $\prod_{\alpha\in \Delta} X_{\alpha}$ are $\Delta$-tuples.

Also, when $\Delta$ is uncountable, it's not that $(x_{\alpha})$ can't be written out as a set of coordinates, but that $(x_{\alpha})$ can't be written in a list of coordinates.

Everything else you have makes sense.

 

[Deveno]

I was wondering, Euge, if for Peter's benefit, you might make explicit the hom-set isomorphisms involved when $A = \Bbb N$ and the field is $\Bbb R$, as well as each factor space, so that we can identify the the direct sum (as a vector space) with:

$\Bbb R[x]$

and the direct product with:

$\Bbb R[[x]]$.

***********

In general, and very loosely speaking, here, a PRODUCT is something you can "factor", this involves "splitting it up" in some fashion.

A CO-PRODUCT is something you "mash up together" from disparate bits. It's very much like a "free structure" in this regard. The reason (in vector spaces) why the direct sum is so *similar* to the direct product is because vector spaces are already "free objects" (to be precise, free $F$-modules).

Much of this "niceness" is preserved by certain categories, called Abelian categories. The proto-typical abelian category (the one whose properties we are trying to generalize, as much as we can) is the category of Abelian groups, which in turn derive much of THEIR niceness from the integers. Many of the theorems you encounter in various guises, are general theorems of Abelian categories, and these are the natural setting for exploring algebraic topology (in particular co-homology).

 

[Math Amateur]

I was wondering, Euge, if for Peter's benefit, you might make explicit the hom-set isomorphisms involved when $A = \Bbb N$ and the field is $\Bbb R$, as well as each factor space, so that we can identify the the direct sum (as a vector space) with:

$\Bbb R[x]$

and the direct product with:

$\Bbb R[[x]]$.

***********

In general, and very loosely speaking, here, a PRODUCT is something you can "factor", this involves "splitting it up" in some fashion.

A CO-PRODUCT is something you "mash up together" from disparate bits. It's very much like a "free structure" in this regard. The reason (in vector spaces) why the direct sum is so *similar* to the direct product is because vector spaces are already "free objects" (to be precise, free $F$-modules).

Much of this "niceness" is preserved by certain categories, called Abelian categories. The proto-typical abelian category (the one whose properties we are trying to generalize, as much as we can) is the category of Abelian groups, which in turn derive much of THEIR niceness from the integers. Many of the theorems you encounter in various guises, are general theorems of Abelian categories, and these are the natural setting for exploring algebraic topology (in particular co-homology).

Sounds fascinating to me ... will be trying to master these concepts/ideas ...

Still somewhat challenged by Euge's examples 2 and 3 ...

But will be working deliberately and carefully through ring and module theory ... and category theory as necessary as well ...

Thanks to you and Euge for your generous help and insights ...

Peter

 

[Euge]

I was wondering, Euge, if for Peter's benefit, you might make explicit the hom-set isomorphisms involved when $A = \Bbb N$ and the field is $\Bbb R$, as well as each factor space, so that we can identify the the direct sum (as a vector space) with:

$\Bbb R[x]$

and the direct product with:

$\Bbb R[[x]]$.

***********

Well, I intend to use the isomorphisms in Example 2 to identify certain power series rings in two different ways. The spaces $R[x]$ and $R[[x]]$ can be naturally identified with $R_0$ (the space of eventually zero sequences of real numbers) and $R^\infty$ (in Example 1), respectively, via the mapping $x \mapsto (0,1,0,0,\ldots)$, extended linearly.

With these identifications, I can apply the results in Example 2 to show that

$\displaystyle \text{Hom}_{\Bbb R}(R[x], F) \cong F[[x]] \cong \text{Hom}_{\Bbb R}(F, R[[x]])$

whenever $F$ is a finite dimensional $\Bbb R$-algebra. Indeed, if $F$ is a finite dimensional $\Bbb R$-algebra, there are vector space isomorphisms

$\displaystyle \text{Hom}_{\Bbb R}(R[x], F) \cong \prod_{n = 1}^\infty \text{Hom}_{\Bbb R}(\Bbb R, F) \cong \prod_{n = 1}^\infty F \cong F[[x]]$

and

$\displaystyle \text{Hom}_{\Bbb R}(F, R[[x]]) \cong \prod_{n = 1}^\infty \text{Hom}_{\Bbb R}(F, \Bbb R) \cong \prod_{n = 1}^\infty F \cong F[[x]]$.

Without the finite dimensionality hypothesis, $\text{Hom}_{\Bbb R}(R[x], F)$ is identified with $F^*[[x]]$, where $F^*$ is the dual of $F$.

 

[Euge]

Sounds fascinating to me ... will be trying to master these concepts/ideas ...

Still somewhat challenged by Euge's examples 2 and 3 ...

But will be working deliberately and carefully through ring and module theory ... and category theory as necessary as well ...

Thanks to you and Euge for your generous help and insights ...

Peter

Sorry Peter, I didn't mean to give difficult examples. Examples 2 and 3 were meant to be illustrations of how of products and sums are used in different areas of mathematics.