Ring Theory texts and "right notation" for maps/functions

[Math Amateur]

I would like members views on right notation for maps/functions.

I am thinking of studying some material in some of the chapters of the book:

Introduction to Ring Theory by P. M. Cohn

Cohn claims his book is suitable for 2nd and 3rd year undergraduates and the book seems to have some really interesting material in it - for example:

Chapter 2 is on linear algebras and Artinian rings

Chapter 3 is on Noetherian rings

Thus Cohn seems to have really interesting material presented at a level for senior undergraduates.

However I note that the author uses right notations for mappings/functions writing a function as \(\displaystyle x \to xf \)

I am totally unfamiliar with this notation.

Do members think that this notation is any reason to avoid this book, which otherwise looks really interesting?

I would be most interested in members views on this matter.

Peter

 

[Sudharaka]

I would like members views on right notation for maps/functions.

I am thinking of studying some material in some of the chapters of the book:

Introduction to Ring Theory by P. M. Cohn

Cohn claims his book is suitable for 2nd and 3rd year undergraduates and the book seems to have some really interesting material in it - for example:

Chapter 2 is on linear algebras and Artinian rings

Chapter 3 is on Noetherian rings

Thus Cohn seems to have really interesting material presented at a level for senior undergraduates.

However I note that the author uses right notations for mappings/functions writing a function as \(\displaystyle x \to xf \)

I am totally unfamiliar with this notation.

Do members think that this notation is any reason to avoid this book, which otherwise looks really interesting?

I would be most interested in members views on this matter.

Peter

Hi Peter, :)

I moved this to Linear Algebra sub forum since it seems to me as a math related discussion.

In the book I refer, A Course in Ring Theory by Passman both notations are used. I don't know why he opted to use both notations rather than sticking to one notation, but in the first chapter he says,

"We will freely use both right and left function notation throughout this book. The particular choice will always be clear from context."

It would be interesting if someone could come up with a situation in Ring Theory where use of both notations will be beneficial.

 

[Math Amateur]

Hi Peter, :)

I moved this to Linear Algebra sub forum since it seems to me as a math related discussion.

In the book I refer, A Course in Ring Theory by Passman both notations are used. I don't know why he opted to use both notations rather than sticking to one notation, but in the first chapter he says,

"We will freely use both right and left function notation throughout this book. The particular choice will always be clear from context."

It would be interesting if someone could come up with a situation in Ring Theory where use of both notations will be beneficial.

Thanks Sudharaka

Reading between the lines it seems you do not approve of both notations being used ... since presumably you do not see the benefits of this ...

Do you favour the more traditional left notation ... where the image of x under f is written fx or f(x) ...

Peter

 

[Deveno]

"Right notation" allows one to parse composition of maps like one reads, from left to right. Sometimes with "left notation" one must phrase things in terms of "anti-homomorphisms" or "opposite rings" which can be inconvenient.

Ideals, and module actions are often "one-sided", and sometimes "one side" fits better with "other objects", it depends on how you are "combining things together".

An example: in groups, it is customary to write the conjugate of $a$ by $g$ as:

$gag^{-1}$.

The trouble with doing this, is that it makes conjugation an "anti-homomorphism", if we write $a^g = gag^{-1}$, then:

$(a^g)^h = a^{hg}$

If, however, we "hit $a$ with $g$ from the RIGHT", we get:

$(a^g)^h = a^{gh}$ which seems like the "proper way" to do things.

A lot of this only applies in non-commutative situations, however, these are often the most interesting.

 

[Math Amateur]

"Right notation" allows one to parse composition of maps like one reads, from left to right. Sometimes with "left notation" one must phrase things in terms of "anti-homomorphisms" or "opposite rings" which can be inconvenient.

Ideals, and module actions are often "one-sided", and sometimes "one side" fits better with "other objects", it depends on how you are "combining things together".

An example: in groups, it is customary to write the conjugate of $a$ by $g$ as:

$gag^{-1}$.

The trouble with doing this, is that it makes conjugation an "anti-homomorphism", if we write $a^g = gag^{-1}$, then:

$(a^g)^h = a^{hg}$

If, however, we "hit $a$ with $g$ from the RIGHT", we get:

$(a^g)^h = a^{gh}$ which seems like the "proper way" to do things.

A lot of this only applies in non-commutative situations, however, these are often the most interesting.

Thanks Deveno ... so indeed there are benefits to right notation ...

So I take it you would not be deterred or put off by Cohn's use of right notation in his book on ring theory ...

Peter

 

[Opalg]

In analysis the "left"notation is universal, partly for historical reasons. The classical functions of analysis have always been written that way, and as far as I know nobody has ever tried to write $x\log$ or $\theta\cos$. But in algebra the emphasis is sometimes on a set that is being acted on by a function, rather than a function that is acting on a set. If for example you are looking at a set $X$ that is acted on by a group $G$, it might seem more natural to write the action as $xg$ or $x^g$ rather than $g(x)$.

 

[Deveno]

Thanks Deveno ... so indeed there are benefits to right notation ...

So I take it you would not be deterred or put off by Cohn's use of right notation in his book on ring theory ...

Peter

My first algebra book was Herstein's Topics In Algebra, and he used "right notation". It was very strange, at first.

But then, like those people who wear eyeglasses that invert images upside-down, all of a sudden, it stopped being "backwards" and became natural.

Notation is a tool, the ideas we describe with our tools are the main thing. As long as there is an "agreement" of which convention is being used, I don't think it matters.

In fact, it's good to think "dually" and try to imagine the "mirror image" of a "sided" statement. This applies to many more things than just mathematics.