- #1

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I am reading Dummit and Foote's book "Abstract Algebra" (3rd Edition) and am focused on Chapter 15: Commutative Rings and Algebraic Geometry ... ...

On page 657 D&F give a definition of a k-algebra ... as follows:

I have to say I do not find that this definition gives me a good intuitive idea of the nature of an algebra over a field ... ...

I much prefer the definition given by Cooperstein in his book "Advanced Linear Algebra" (Second Edition) where he defines what he calls an "associative algebra over a field F" (which I take to be the same as D&F's k-algebra ... ... is that right?) ... ...

Cooperstein's definition is as follows:

I find Cooperstein's definition more clear regarding the nature of an algebra ... ... but I am currently working from D&F and wish to fully understand the approach to an algebra ... but I am unclear on exactly why these two definitions are the same or equivalent ... ...

Can someone please help me to formally and rigorously prove that these two definitions are equivalent ... ...

Help will be much appreciated ...

Peter

On page 657 D&F give a definition of a k-algebra ... as follows:

I have to say I do not find that this definition gives me a good intuitive idea of the nature of an algebra over a field ... ...

I much prefer the definition given by Cooperstein in his book "Advanced Linear Algebra" (Second Edition) where he defines what he calls an "associative algebra over a field F" (which I take to be the same as D&F's k-algebra ... ... is that right?) ... ...

Cooperstein's definition is as follows:

I find Cooperstein's definition more clear regarding the nature of an algebra ... ... but I am currently working from D&F and wish to fully understand the approach to an algebra ... but I am unclear on exactly why these two definitions are the same or equivalent ... ...

Can someone please help me to formally and rigorously prove that these two definitions are equivalent ... ...

Help will be much appreciated ...

Peter