Significance of difference between ring and algebra

[algebrat]

(MAJOR EDIT: I think I missed the associative part, is that more or less my only mistake?)

I've got an un"well-formed" question, I've been staring at things like every ring is a module over itself, counting the number of sets and operations in various definitions of algebraic objects.

I was looking at definition of algebra (over comm. ring), and thinking, "what's the point, it looks like a ring, what's the difference"?

So what I just decided, and I'm not sure if this is best way to chop it up, is that:

1. every algebra is a ring
2. every ring is an algebra
3. every noncommutative ring is an algebra over it's center, a comm. ring

Is this correct? So algebras coincide with interest in noncommutative rings, and really just the noncommutative ring looked at as a module over it's center, the commutative ring part.

If we look at the module over some units, then it is a vector space with a product for vectors, or an algebra over a field, or a noncommutative ring of a special type.
Is this a decent/good/normal way to see these topics?

 

[mathwonk]

an algebra is a ring MAP, i.e. a ring is one ring, an algebra is two rings and a ring map between them.

 

[DonAntonio]

an algebra is a ring MAP, i.e. a ring is one ring, an algebra is two rings and a ring map between them.

That's a rather succinct and, imo, incomplete definition: not every map between rings makes the range ring into an algebra over the domain ring as there must be some compatibility between the different operations.

For starters, let's say a (very important, basic kind of) algebra is a ring which is also a vectorial space over some field, in such a way that the different operations "behave" correctly when mingling. For example, a polynomial ring over a field is an algebra over that field, the set of all square matrices over a field is an algebra over that field. These two examples are, imo and by far, the two most important ones for most basic uses.

DonAntonio

 

[micromass]

That's a rather succinct and, imo, incomplete definition: not every map between rings makes the range ring into an algebra over the domain ring as there must be some compatibility between the different operations.

For starters, let's say a (very important, basic kind of) algebra is a ring which is also a vectorial space over some field, in such a way that the different operations "behave" correctly when mingling. For example, a polynomial ring over a field is an algebra over that field, the set of all square matrices over a field is an algebra over that field. These two examples are, imo and by far, the two most important ones for most basic uses.

DonAntonio

You're saying the same thing as mathwonk (but mathwonk apparently restricts himself to commutative rings). An algebra over a ring R is a ring A with a (unital) ring homomorphism [itex]\alpha:R\rightarrow A[/itex]. (if the rings are not commutative, then we demand that the image of that homomorphism is in the center of A).
We can then define a multiplication by

[tex]r\cdot a=\alpha(r)a[/tex]

 

[DonAntonio]

You're saying the same thing as mathwonk (but mathwonk apparently restricts himself to commutative rings). An algebra over a ring R is a ring A with a (unital) ring homomorphism [itex]\alpha:R\rightarrow A[/itex]. (if the rings are not commutative, then we demand that the image of that homomorphism is in the center of A).
We can then define a multiplication by

[tex]r\cdot a=\alpha(r)a[/tex]

Well, I mentioned the compatibility between different operation, which is very important imo. The OP will now go and study about that, or not. You're definition, which you way mathwonk gave, even requires the rings to be unital (and thus, apparently, the homom's to map the unity into the unity)

The definition you're giving is from scratch, which would perhaps look a little harsh for a beginner. I'd rather try to introduce someone into the subject by using well known examples and checking the operations there behave nicely. A matter of taste, perhaps.

DonAntonio

 

[mathwonk]

thanks for the clarifications. to be more precise, in the case of commutative rings, an algebra is just a (unital) ring map.

More generally, as micromass says, an algebra is a (unital) ring map whose image is in the center of the target.

The desired compatibility of operations is included in the definition of "ring map".

Both DonAntonio's standard examples are included here.

more generally an A algebra (where A is commutative) is an A module E plus an A bilinear map ExE-->E.One may prefer other introductions, but this one is easy and memorable. No one will ever forget it. I agree the two basic examples are very illustrative. I suggest that with this definition it is a trivial exercise even for a beginner to verify that both are algebras.

 

[algebrat]

Hmm, everyone seems to be diverging away from my emphasis on noncommutative rings, in that they like polynomials as an example. I guess I'll budge a little and think of it, algebra, as two sets again, modules being sort of the basic example (of a two set "algebra") in advanced algebra, and where the abelian group over the rings is in fact a ring. (I used algebra three different ways in a sentence.)

But I'll keep in mind that algebras treat some classic noncommutative rings.

And then group ring [STRIKE]is this annoying thing that seems to nicely absorb polynomials and adjuncts[/STRIKE], but I haven't decided where I should organize it, is it almost an algebra, but a special case. Is it sort of any algebra with the point of view of a basis. Oh, where the basis vectors for the algebra are invertible, there, so a group ring looks like a special case of an algebra. (I'm ignoring delicacies like associative and commutative parts blah blah.)

 

[mathwonk]

as you probably know, there are also non commutative polynomial rings. e.g. the ring of matrices with entries chosen from a polynomial algebra over field, can be regarded as a ring of polynomials with coefficients in the ring of matrices over the field. this seems to be a non commutative algebra over the commutative ring of polynomials with coefficients in the field. this is one nice way to prove the cayley hamilton theorem.

 

[algebrat]

as you probably know, there are also non commutative polynomial rings. e.g. the ring of matrices with entries chosen from a polynomial algebra over field, can be regarded as a ring of polynomials with coefficients in the ring of matrices over the field. this seems to be a non commutative algebra over the commutative ring of polynomials with coefficients in the field. this is one nice way to prove the cayley hamilton theorem.

Ah, beautiful, thank you for pulling my thoughts together to something new. I will think about this soon! I had heard of the polynomials over matrices definitely, but I had not remembered them in my recent thoughts/framework.

By recent "thoughts/framework" I mean group rings are an example of algebra and module, which are each an example of two sets each with some operations. Then groups and rings are single sets, one with one operation, the other with two operations. I haven't gotten around to deciding how to contrast group rings with these other algebras and modules just having two sets, maybe the invertible basis elements as I offered above.

 

[mathwonk]

note this is closely related to one of DonAntonio's favorite examples, i.e. matrices over a commutative ring of polynomials.

in another guise, it is a graded version of matrices over a field. i.e. take an infinite direct sum of that ring graded by the non negative integers. then multiply so that matrices of grade n times matrices of grade n wind up in grade n+m.

then you have the polynomials with matrix coefficients again, and you might regard them as a sort of algebra over the non commutative ring of matrices over a field, but i haven't thought about the properties or problems the non commutativity will give you.

 

[algebrat]

.. I had heard of the polynomials over matrices definitely, but I had not remembered them in my recent thoughts/framework...

Oh, I misunderstood, I have some careful reading to do, i.e., I read polynomials over matrices, when you said, matrices over polynomials.

But I just looked up Cayley-Hamilton as you mentioned, which when I look up the wikipedia page on that, I would look at the machinery as polynomials over matrices. Is there some intersect in two definitions?

http://en.wikipedia.org/wiki/Cayley-Hamilton_theorem

EDIT: hmm, they mention this "Since the entries of the matrix are (linear or constant) polynomials in λ", so perhaps there is some intersect going on. Something like, the polynomial over a matrix is a matrix over a polynomial.

But today I am off to read about homology in Lang's Algebra, so I'll have to come back later and read this thread with pen and paper.

 

[mathwonk]

here are some notes for you from one of my courses. wikipedia also has this proof it seems, (see 3.3)

i learned it from a. adrian Albert's book fundamental concepts of higher algebra,

http://www.abebooks.com/servlet/SearchResults?an=a.+adrian+Albert&sts=tthis book was published in 1956, sold for $1.35 in 1959, and treats polynomials over non commutative rings, on pages 37-40.

but the wikipedia article may be clear as well. i haven't read it but they seem to discuss the concepts of left and right substitution, which are key."Oh, I misunderstood, I have some careful reading to do, i.e., I read polynomials over matrices, when you said, matrices over polynomials. " mirabile dictu, it does not matter! in fact i said both.

the whole point is there are two ways to look at the same algebra, as matrices with polynomial entries, or as polynomials with matrix coefficients. you get the same, or rather isomorphic, algebras.